\section{Introduction}
Nowadays web users are confronted with an overabundance of information caused by a constant increase in the volume of information that users can no longer absorb, process, or prioritize. As a result, the selection and use of information processed by users become particularly complex. This phenomenon presents the need to design a recommender system that best depicts the preferences of a user with regard to the most relevant information at possibly the shortest time \cite{ref1,ref2}.\newline
\indent Recommender systems are tools whose purpose is to help users overcome information overload by selecting the most interesting information based on their preferences. In other words, they try to predict a user's interest towards an item. The items to recommend are varied, ranging from movies to watch, books to read, podcast to listen to, or else. Conventional recommender systems have been successfully applied by e-commerce or social networking websites such as Amazon (www.amazon.com), Netflix (www.netflix.com), YouTube (www.youtube.com), and Facebook (www.facebook.com). Since then many recommender system algorithms and their variants have been proposed in literature, however, most of them were mainly accuracy-oriented algorithms that predict the rating of an item. In other words, these algorithms were focused on optimizing the accuracy of the rating of a predicted item. Although such recommender system algorithms are sufficient in many applications, there are situations where they may not be enough because they do not take into account all facets of the user's interests such as the desire to change. At this point the need to design a new paradigm of recommender systems arises considering important factors outside the optimization of the accuracy of the rating of predicted item. \\
\indent A multinomial probabilistic model for movie genre prediction is proposed in this paper as a response to the confronting challenge. Our model aims to predict a movie's genre rather than to predict its rating. Technically, this approach is based on Bayesian reasoning and is enacted in two steps. In the first step, the multinomial probabilistic model is applied to learn the movie's likelihood of belonging to a particular genre using multinomial model. At the second step, the Bayesian probabilistic reasoning is applied for the prediction of the movie's genres. This in turn completes the item's genre given by experts and consequently can improve the recommendation.\\
In the recent past, many recommender systems that consider factors other than the accuracy-oriented ones and their variants have been proposed in literature \cite{ref3,ref4,ref5,ref6}. To the best of our knowledge, our model predicting an item's genre/s that complement the genres assigned by experts is a new attempt in the field of recommender systems. The rest of the paper is organized as follows. Section 2 discusses the related work. Section 3 outlines the proposed algorithm. Section 4 contains the performance study. Finally in section 5, we summarize our work.

\section{Related work}

Traditional recommender systems focus on increasing the accuracy of ratings prediction of items not yet viewed by active users based on ratings of items already viewed by active users. However, in \cite{ref7},  McNee  et  al. stated that focusing only on improving the accuracy of the recommended items' rating is not enough for improving the user satisfaction. Factors such as coverage, diversity, novelty and/or serendipity should be used to improve the recommendations. Aiming at enhancing the conventional recommender systems while working with important factors outside  item  rating  prediction  accuracy,  many studies developing new  paradigm of making recommendation were proposed in the literature. We briefly discuss related works considering fcators outside of accuracy  in recommender systems and survey works that have done new attempts in the area of recommender systems.  \\
\indent Saul et al. \cite{ref8} enhanced the diversity of recommendations by adopting the re-ranking approach with a greedy selection. This approach allows to discover new useful recommendations that were not searched by the user. In \cite{ref9}, Tuzhilin and Adamopoulos proposed a probabilistic neighborhood selection in collaborative filtering that directly generates diversified recommendation lists prior to overcoming the overspecialization issues in traditional recommendations. Recent researches have been focused on designing new attempts in the literature. Liu et al. \cite{ref10} proposed a novel approach of collaborative filtering whose goal is to select a small set of special users called star users. Unlike traditional approaches, collaborative filtering allows finding similar users or neighborhood for each target user. Collaborative filtering based on star users employs star users to represent the interest of the whole set of users prior to computing the recommendation, so as to improve the scalability of the traditional recommender systems. \\
\indent Amatriain et al. \cite{ref11} proposed a variation of the traditional collaborative filtering approach that uses expert opinions from independent data set to compute the recommendations. In this approach, the nearest neighbor-oriented recommendation was scaled-up by the use of these external experts.

\section{The Naive Bayes Classifier for text classification}

In order to solve classification problems, many methods such as linear discriminant analysis, neural networks, support vector machines, etc have been proposed. Among them, one of the oldest and well-known classification algorithms is the Naive Bayes classification algorithm which dates back to 18th century. The algorithm performs well in various tasks despite its simplicity, however it comes with a naive assumption that features that constitutes features vectors are independent. One of the well-known applications of Naive Bayes is the text classification. The details of the Naive Bayes classifier are well summarized in \cite{ref12,ref13}. This section will provide a brief overview of the Naive Bayes classifier applied to text classification.\\
\indent The Naive Bayes classifiers is based on the Bayesian theorem and is appropriate for multidimensional data i.e. multidimensional feature vectors. Regarding the text document classification problem, a document is represented as a bag of words where each word is part of a vocabulary $V$, $ w \in V $. Considering each document $ d \in D $ where $ D $ represents the set of all training documents and every document is labeled as $c$ where $c \in C $ is a set of $\lvert C \rvert$ distinct classes. According to the Bayes theorem the Naive Bayes classifiers estimates the probability of a class $c$ given a document $ d $ as follows:
\begin{equation}
P( c \mid d ) = \frac{P(c) \cdot P( d\mid c )}{P(d)}
\label{eq:1}
\end{equation}
\indent This estimate can be further used for document classification by simply assigning the class label that corresponds to the highest probability $P( c \mid d )$. The probability $P(c)$ is the prior probability of a document being of class $c$ without actually knowing anything about the document.  The prior class probability $P(c)$ is estimated by counting the number of training  documents in every class: \\
\begin{equation}
P(c)=\frac{\sum_{d \in D}P(   c \mid d)}{\lvert D \rvert}
\label{eq:2}
\end{equation}
where $P(c\mid d) \in \{0, 1\}$, is 1 if document is labeled as $c$ otherwise it is 0 and $\lvert D \rvert$  is the total number of training documents.
The conditional probability  $P( d\mid c )$ incorporates information about the document via the set of words that occurred in the document. Depending on a class $c$ of a document, the probability of a word occurrence $P( w\mid c )$ will vary as well as $P(d \mid c )$. To simplify the computation of the $P(d \mid c )$, $d = \{w_{1},w_{2},...,w_{\lvert d \rvert}\}$, where  $\lvert d \rvert$ is the  document  length,  it is assumed that words are conditionally independent, i.e. the occurrence of words do not depend upon each other.  Thus, $P(d \mid c) = P( w_{1}\mid c ) \cdot P( w_{2}\mid c )...P( w_{\mid d \mid }\mid c )$. A slightly different form of computing  $P(d \mid c)$ is given as follows
\begin{equation}
P(d \mid c) =  \prod\limits_{w \in V}P( w\mid c )^{N(w)}
\label{eq:3}
\end{equation}
where $w_{i}$ is a unique word in the vocabulary $V$. Here, instead of iterating over the words in a document we iterate over the set of all words in the vocabulary $V$. Notice that some words from the vocabulary might occur multiple times in that case $1 < N(w)$, and on the other hand some words might not occur in the document at all, in such a case $N(w) = 0$. Substituting eq. \ref{eq:3} to eq. \ref{eq:1} and assuming  $P(d_i) = P(d_j), \forall i,j$, i.e. all documents are equally to occur, the classification can be performed as follows :
\begin{equation}
c = \arg\max_{c} P(c)\prod\limits_{w \in V}P( w\mid c )^{N(w)}
\label{eq:4}
\end{equation}

Probabilities $P( w_\mid c )$ of a word $w$ occurring in a class $c$ is estimated by counting the number of times a word $w$ appeared in all documents of class $c$. However, some words might not appear at all in documents of a particular class resulting in a zero probability. To avoid the zero probability in such cases a small probability is assigned according the Laplace's Law of succession:
\begin{equation}
P( w\mid c)=\frac{1 + N(w,c)}{\lvert V \rvert + \sum_{w \in V} N(w, c)}
\label{eq:5}
\end{equation}
 where $N(w,c) $ represents the number of times the word $w$ occurred in the training documents whose class label is $c$, $N(c)$ denotes the number of documents labeled as $c$. \\

\section{Proposed method}

We propose to employ a similar approach of the text-classification\cite{ref14} discussed in the previous section to predicting movies' genres. Following the approach of bag-of-words for text classification we propose a bag-of-users approach to predict movies' genres. Such prediction is possible due to the fact that users are usually consistent with their preferences, and prefer some genres over others.\newline
\indent In the following, we introduce symbols and definitions used in this section. Let $U$ be the set of all users, $ M $ be the set of all movies and $ G $ be the set of genres. The movie $ m \in M $ is represented by $\lvert U \rvert$ dimensional feature vector $ r_m = \{\mathbf{R}_{u_1,m}, \mathbf{R}_{u_2,m},...,\mathbf{R}_{u_{\vert U \rvert}, m}\}$, where $\mathbf{R}$ is rating matrix and $\mathbf{R}_{u_i,m}$ is either a 0 or a 1, depending if the user $u_i$ rated a movie $m$ or not.\\
\indent It is assumed that users are conditionally independent. That is, the ratings of one user do not depend on ratings of another user. This assumption is required by the Naive Bayes classifier. Thus, the movie likelihood $P( m \mid g )$, can be written as a categorical distribution, that is a particular case of multinomial distribution:

\begin{equation}
P( m \mid g ) \propto \prod\limits_{u \in U}P(u \mid g)^{\mathbf{R}_{u, m}}
\label{eq:6}
\end{equation}

where $\mathbf{R}_{u, m}$ represents whether the user $u$ rated a movie $m$ or not.  We use categorical distribution because a user does not rate the same movie twice, whereas in the text classification the same word can appear in the text more than once. It is often the case that ratings are not binary. There could be more than one rating. For example, 0 corresponds to no rating, 1 to a dislike, 2 to a neutral response and 3 to a like. Then matrix $\mathbf{R}$ is split into three binary matrices $\mathbf{R}_1$, $\mathbf{R}_2$, $\mathbf{R}_3$.  Then an estimate of the probability $P(u \mid g )$ is computed as follow:

\begin{equation}
P( u \mid g ) =\frac{1+ N(u,g)}{\lvert|U \rvert+\sum_{u \in U}N(u,g)}
\label{eq:7}
\end{equation}

This probability is estimated as a relative frequency of user's $ u $ ratings of movies labeled as genre $ g $ to the total number of users' ratings of the movies from that genre. \newline
\indent Note that the Laplace’s Law of succession was used to avoid the zero probability estimates that can occur for situations where a user $ u $ did not rate a movie $m$.

Giving a movie training set $ M $,  the prior probability of $g$ can be estimated as the relative frequency of movies of genre $g$ to the total number of movies:

\begin{equation}
P(g)=\frac{\sum_{m \in M}P(g \mid m)}{\lvert M \rvert}
\label{eq:8}
\end{equation}
where $ P(g \mid m) $ takes the value of $1$ if $m$ is marked as genre $g$, $0$ otherwise. In the case when a number of $N$ genres are assigned to a movie, the probability is corrected as follows $ P(g \mid m) = 1/N$.    \newline

Figure 1 shows the preferences of all of the users in the training set calculated using eq. \ref{eq:7}. Figure 2 illustrates  the conditional probability calculated according to eq. \ref{eq:6} with prior taking the logarithm.

\begin{figure}[h]
\centering
\includegraphics[trim=3.5cm 8cm 4cm 7.5cm, width=0.4\textwidth]{images/figure1.pdf}
\caption{The user preferences represented by $P( u \mid g )$.}
\end{figure}

\begin{figure}[h]
\centering
\includegraphics[trim=3.5cm 8cm 4cm 7.5cm, width=0.4\textwidth]{images/figure2.pdf}
\caption{The logarithm of conditional probabilities $P( m \mid g )$.}
\end{figure}

\section{Experimental evaluation and discussion}

To evaluate the effectiveness of the proposed methodology of predicting movie genres using users’ ratings, we employed the 1M Movielens dataset. The dataset contains one million ratings of 3952 movies given by 6040 users. Each user rated at least 20 movies on the scale from 1 to 5, i.e. dislike, slightly dislike, neutral, slightly like and like. A rating of 1 shows how strongly a user dislikes a movie while a rating of 5 shows how strongly a user likes a movie. The experiments were carried out in the following procedure: a portion of the movies was randomly selected for training and the remaining movies were used to assess the quality of genre prediction. The portion of the movies used for training varied from 5\% to 95\% with a 5\% step, so overall we experimented with 19 different training sizes. For every portion we repeated the experiment 50 times, then estimated the mean and the standard deviation. Among all movies about 33\% were assigned two genres, 10\% three genres, and less than 3\% four, five and six genres, and the remaining 51\% of the movies were labeled with only one genre. We considered the prediction is successful if the predicted genre matches the true one, or one of the true ones if a movie simultaneously was assigned to a number of genres. Prediction was performed separately for every rating. Figure 3 summarizes the evolution of genre prediction.

\begin{figure}[h]
\centering
\includegraphics[trim=4.5cm 10cm 5cm 9cm, width=0.4\textwidth]{images/figure3.pdf}
\caption{Genres' prediction rate depending on the size of the training set.}
\end{figure}

As it can be seen in the figures, the more movies are used in the training the better prediction is. However, an interesting observation here is that even 5\% of the data used for training is enough to obtain 62\% prediction rate. If the prediction was performed simply by using the prior probability, only 30\% of the movies would be predicted correctly. Increasing the training set does improve the prediction rate however, the improvement is no as significant as for the first 5\%. For instance, increasing the size of the training set to 95\% would only improve the prediction rate by a slightly more than 15\%.  From this analysis we are ought to conclude that even a small portion of information about the users' ratings is enough to learn users’ preferences that allows us to predict genres. Particularly, the multinomial model was capable of predicting movies' genres based on users' preference profiles well enough.
The second interesting result is that better predictions are achieved for neutral responses i.e. movies with ratings 3 and for slightly likes i.e. movies with ratings 4, rather than for the expected rating 5. The worst predictions are based on the rating 1 and 2. These results might be ambiguous, however after counting the number of times ratings were given in every rating category, we found that the most popular ratings are 3 and 4 (fig. 4). Thus, the higher number of ratings provides higher prediction accuracy.
Currently, the prediction was performed separately for every class of ratings, for instance, only based on strong dislikes. As for the further research we are planning to investigate if some combinations of ratings will improve the prediction.

\begin{figure}[h]
\centering
\includegraphics[trim=4.5cm 10cm 5cm 9cm, width=0.4\textwidth]{images/figure4.pdf}
\caption{Number of ratings of each type given in the whole set.}
\end{figure}

\section{Conclusion and future work}

Recommender systems are tools whose purpose is to help users overcome information overload by selecting the most interesting information based on users' preferences and also to help companies sell more products. In this paper, we proposed to predict a movie’s genre rather than predicting movie’s ratings. For this purpose, we applied Naïve Bayes classifier and multinomial event model, that previously was used for text classification and a number of other tasks. This approach can potentially improve the efficiency of recommendation by recommending movies from unexpected but relevant genres, this will be investigated in further research. We showed that the genre prediction rate increases while the size of the training data increases, however the rate of prediction slows down. Even a small number of users allows significantly increase genre prediction. That is, overall, we successfully applied ratings for genre prediction in the context of movies.

\ifCLASSOPTIONcaptionsoff
  \newpage
\fi


\section{Introduction}

Let $\mathbb R[X]:=\mathbb R[X_1,\cdots,X_n]$ be the ring of polynomials in the variables $X_1,\cdots,X_n$ with real coefficients.  Denote by $\Delta_n$ the standard $n$-simplex in $\mathbb R^n$, which is defined by
$$ \Delta_n:=\{x=(x_1,\cdots,x_n)\in \mathbb R^n| x_i\geq 0, \sum_{i=1}^n x_i=1\}.$$

P\'{o}lya \cite{Po} proved in 1928 that for a homogeneous  polynomial $f\in \mathbb R[X]$, \textit{if $f(x)>0$ for every $x\in \Delta_n$, then there exists a sufficiently large number $N$ such that all coefficients of the polynomial $(X_1+\cdots+X_n)^N \cdot f$ are positive}.

Powers and Reznick \cite{PR}   gave  an explicit bound for  the number $N$, and applied it to give a constructive version of Handelman's Positivstellensatz.  More explicitly, let $P\subseteq \mathbb R^n$ be a convex, compact polyhedron with non-empty interior, bounded by linear polynomials $L_1, \cdots, L_m\in \mathbb R[X]$. By choosing the sign of the $L_i$'s, we may assume that
\begin{equation}\label{polyhedron}
P=\{x\in \mathbb R^n| L_i(x)\geq 0, ~ i=1,\cdots,m\}.
\end{equation}

\begin{theorem}[{Handelman's  Positivstellensatz, \cite{Ha}}] \label{thr-Handelman} For a polynomial $f\in \mathbb R[X]$, if $f(x)>0$ for all $x\in P$, then $f$ can be represented as
$$f = \sum_{|\alpha|\leq M} f_\alpha L_1^{\alpha_1}\cdots L_m^{\alpha_m} $$
 for some $M\in \mathbb N$ and $f_\alpha\geq 0$ for all $\alpha=(\alpha_1,\cdots,\alpha_m) \in \mathbb N^m$ such that $|\alpha|\leq M$.
\end{theorem}
 J.L. Krivine \cite{K}   proved Handelman's Positivstellensatz for a special polyhedron. Moreover, one can find a   generalization of this Positivstellensatz in \cite[Theorem 5.4.6]{PreD} (or \cite[Theorem 7.1.6]{M}).  A bound for the number $M$  was   given by Powers and Reznick \cite{PR}, using the bound for the number $N$ in P\'{o}lya's Positivstellensatz.

Theorem \ref{thr-Handelman} yields the following consequence.

\begin{corollary} For a polynomial $f\in \mathbb R[X]$, if $f(x)>0$ for all $x\in P$, then $f$ can be represented as
$$f = \sum_{e=(e_1,\cdots,e_m)\in \{0,1\}^m}  f^2_e L_1^{e_1}\cdots L_m^{e_m}, $$
 where   $f_e\in \mathbb R[X]$  and $\deg(f_e^2) \leq M$.
\end{corollary}

This corollary is  a special case of Schm\"{u}dgen's Positivstellensatz  \cite{Schm91} for convex, compact polyhedra  which includes an explicit bound on the degrees of sums of  squares coefficients $f_e^2$.

Schm\"{u}dgen's Positivstellensatz has many important applications, especially in solving polynomial optimization problems and moment problems for compact semi-algebraic sets. Therefore, as a special case of Schm\"{u}dgen's Positivstellensatz,   Handelman's theorem for polynomials plays an important role in application.

  A matrix version of  P\'{o}lya's Positivstellensatz was given by Scherer and Hol \cite{SchH}, with  applications e.g. in robust polynomial semi-definite programs.  Schm\"udgen's theorem for operator polynomials has been discovered  by Cimprič and Zalar \cite{CZ}. Positivstellens\"atze for polynomial matrices have been studied by some other authors, see for example in \cite{Ci11}, \cite{Ci12}, \cite{SaSch}, \cite{Le}.     \textit{The main aim of this paper is to give a version of Handelman's Positivstellensatz for polynomial matrices with an explicit degree bound. }

We need to introduce some notations.  For $t\in \mathbb N^*$, let  $\mathcal{M}_t(R)$  denote the ring of square matrices of order $t$ with entries from  a commutative unital ring  $R$. Denote by $\mathcal{S}_t(R)$ the subset of  $\mathcal{M}_t(R)$  consisting of all symmetric matrices. \\
In this paper we consider mainly $R$ to be the   ring  $\mathbb R[X]$   of polynomials in $n$ variables $X_1,\cdots, X_n$ with real coefficients.  Each element $\bold{A} \in  \mathcal{M}_t(\mathbb R[X])$ is a matrix whose entries are polynomials from $\mathbb R[X]$, called a \emph{polynomial matrix}. Each element $\bold{A}\in \mathcal{M}_t(\mathbb R[X])$ is also called a \emph{matrix polynomial},  because it can be viewed as a polynomial in $X_1,\cdots, X_n$ whose entries from $\mathcal{M}_t(\mathbb R)$. Namely, we can write $\bold{A}$ as
$$  \bold{A}=\sum_{|\alpha|=0}^{d} \bold{A}_{\alpha}X^{\alpha}, $$
where $\alpha=(\alpha_1,\cdots,\alpha_n) \in \mathbb N^{n}$, $|\alpha|:=\alpha_1+\cdots + \alpha_n$, $X^{\alpha}:=X_1^{\alpha_1}\cdots X_n^{\alpha_n}$, $\bold{A}_\alpha \in \mathcal{M}_t(\mathbb R)$, $d$ is the maximum over all degree of entries of $\bold{A}$, and  it is called the \textit{degree of the matrix polynomial $\bold{A}$}.  To unify notation, throughout the paper each element of $\mathcal{M}_t(\mathbb R[X])$ is called a \emph{polynomial matrix}. \\
For any polynomial matrix $\bold{A}\in \mathcal{M}_t(\mathbb R[X])$ and for any subset $K\subseteq \mathbb R^n$,  by  $\bold{A}\succcurlyeq 0 $ (resp. $\bold{A}\succ 0$)  \textit{on $K$} we mean that for any $x\in K$, the matrix  $\bold{A}(x)$ is \emph{positive semidefinite} (resp. \textit{positive definite}),  i.e.   all eigenvalues of the matrix $\bold{A}(x)$ are non-negative (resp. positive).   \\
 For any polynomial matrices $\bold{A}, \bold{B}\in \mathcal{M}_t(\mathbb R[X])$, the notation \textit{$\bold{A}\succcurlyeq  \bold{B}$ on $K$} means that $\bold{A}-\bold{B}\succcurlyeq  \bold{0}$ on $K$.

Suppose that we have a convex, compact polyhedron $P\subseteq \mathbb R^n$  with non-empty interior, bounded by linear polynomials $L_1, \cdots, L_m\in \mathbb R[X]$, defined by (\ref{polyhedron}). Let $\bold{F} \in \mathcal{S}_t(\mathbb R[X])$ be a  polynomial matrix of degree $d>0$. Assume $\bold{F} \succ \bold{0}$ on $P$. The main result of this paper is presented in     Theorem \ref{thr:main} which is a  matrix version of Handelman's Positivstellensatz, stating  that there exists a  number $N_0$  such that for all integer $N >N_0$ the polynomial matrix $\bold{F}$ can be written as
$$\bold{F}=\sum_{|\alpha|=N+d}\bold{F}_\alpha L_1^{\alpha_1}\cdots L_m^{\alpha_m},$$
where  $\bold{F}_\alpha \in \mathcal{S}_t(\mathbb R)$ are positive definite scalar  matrices  with $|\alpha|=N+d$.\\
 The main idea  in the proof of this theorem inherits  from Powers and Reznick   \cite{PR}, using a matrix version of P\'{o}lya's Positivstellensatz  \cite{SchH} and the continuity of  eigenvalue functions of the polynomial matrix $\bold{F}$ on the entries  of $\bold{F}$ (by \cite[Theorem 1]{Z}).
  As a corollary of this theorem, we give a special case of Schm\"udgen's Positivstellensatz for polynomial matrices positive definite on  convex, compact polyhedra.  Furthermore, we give a procedure to find such a representation for the polynomial matrix $\bold{F}$.
 \section{Representation of polynomial matrices positive definite on simplices}
In this section we consider a simple case  where $P$ is an $n$-simplex with vertices
$\{v_0,v_1,\cdots,v_n\}$ and let $\{L_0,L_1,\cdots,L_n\}$ be the set of barycentric coordinates of $P$, that is, each $L_i\in \mathbb R[X]$ is linear and
\begin{equation}\label{simplex}
X=\sum_{i=0}^n L_i(X) v_i, ~ \sum_{i=0}^nL_i(X)=1, L_i(v_j)=\delta_{ij}.
\end{equation}
Let $\bold{F}\in \mathcal{S}_t(\mathbb R[X])$ be a polynomial matrix of degree $d>0$. We can express   $\bold{F}$ as
$$\bold{F}(X)=\sum_{|\alpha|\leq d} \bold{A}_\alpha X^{\alpha}, $$
where $\bold{A}_\alpha \in \mathcal{M}_t(\mathbb R)$.

Let us consider the \textit{Bernstein-B\'{e}zier form} of $\bold{F}$ with respect to $P$:
\begin{equation}\label{Bernstein-Bezier}
\bold{\widetilde{F}}_d(Y):=\bold{\widetilde{F}}_d(Y_0,\cdots,Y_n):=\sum_{|\alpha|\leq d} \bold{A}_\alpha \Big(\sum_{i=0}^n Y_iv_i\Big)^\alpha \Big(\sum_{i=0}^n Y_i\Big)^{d-|\alpha|}.
\end{equation}
It is easy to see that $\bold{\widetilde{F}}_d(Y)\in \mathcal{S}_t(\mathbb R[Y])$ is a homogeneous polynomial matrix of degree $d$. Moreover, it follows from the relations (\ref{simplex}) that
$$ \bold{\widetilde{F}}_d(L_0,\cdots,L_n)=\bold{F}(X).$$
Following Scherer and Hol \cite{SchH}, for each multi-index $\alpha=(\alpha_1,\cdots,\alpha_n)\in \mathbb N^n$, let us  denote
$$  \alpha!:=\alpha_1!\cdots \alpha_n!; ~ D_\alpha:=\partial_1^{\alpha_1}\cdots\partial_n^{\alpha_n}.$$
With these notations,  we can re-write $\bold{F}$ as
$$\bold{F}(X)= \sum_{|\alpha|\leq d} \dfrac{D_\alpha \bold{F}(0)}{\alpha!}X^\alpha.$$
With the spectral norm $\|\bold{\cdot}\|$, following Scherer and Hol \cite{SchH}, we define
\begin{equation}\label{pt:norm}
C(\bold{F}):=\max_{|\alpha|\leq d}\dfrac{\|D_\alpha \bold{F}(0)\|}{|\alpha|!}.
\end{equation}

Using these notations,  we have the following representation of polynomial matrices which are positive on simplices.
\begin{theorem} \label{thr-simplex} Let $P\subseteq \mathbb R^n$ be an $n$-simplex given as above and $\bold{F}\in \mathcal{S}_t(\mathbb R[X])$ a polynomial matrix of degree $d>0$. Assume that $\bold{F}\succcurlyeq \lambda \bold{I}_t$ on $P$ for some $\lambda >0$.  Let  $C:=C(\bold{\widetilde{F}_d})$. Then for each $N>\dfrac{d(d-1)}{2} \dfrac{C}{\lambda}-d$, $\bold{F}$ can be represented as
$$\bold{F}=\sum_{|\alpha|=N+d}\bold{F}_\alpha L_0^{\alpha_0}\cdots L_n^{\alpha_n},$$
where  each  $\bold{F}_\alpha\in \mathcal{S}_t(\mathbb R)$ is positive  definite.
\end{theorem}

\begin{proof} Let us denote by $\Delta_{n+1}$ the standard simplex in $\mathbb R^{n+1}$, i.e.
$$ \Delta_{n+1}=\{(y_0,\cdots,y_n)\in \mathbb R^{n+1}| y_i\geq 0, \sum_{i=0}^n y_i=1\}. $$
Since $\bold{F}(x)\succcurlyeq \lambda \bold{I}_t$  for all $x\in P$, the  Bernstein-B\'{e}zier form
 $\bold{\widetilde{F}_d}$ of $\bold{F}$ with respect to $P$ satisfies
$$ \bold{\widetilde{F}_d}(y_0,\cdots,y_n)\succcurlyeq \lambda \bold{I}_t, \forall (y_0,\cdots,y_n)\in \Delta_{n+1}. $$
Then it follows from P\'{o}lya's theorem for polynomial matrices \cite[Theorem 3]{SchH}, that for each $N> \dfrac{d(d-1)}{2} \dfrac{C}{\lambda}-d$,
\begin{equation}\label{pt-cm-simplex}
(\sum_{i=0}^n Y_i)^N\bold{\widetilde{F}_d}(Y)=\sum_{|\alpha|=N+d}\bold{F}_\alpha Y_0^{\alpha_0}\cdots Y_n^{\alpha_n},
\end{equation}
where  each  $\bold{F}_\alpha\in \mathcal{S}_t(\mathbb R)$ is positive  definite. Substituting $Y_i$ by $L_i$ on both sides of (\ref{pt-cm-simplex}), noting that
$$\bold{\widetilde{F}_d}(L_0(X),\cdots,L_n(X)) = F(X) \mbox{ and } \sum_{i=0}^N L_i(X)=1,$$
 we obtain the required representation for $\bold{F}$.
\end{proof}
\section{Representation of polynomial matrices positive definite on convex, compact polyhedra}
Throughout this section, let $P\subseteq \mathbb R^n$ be a  convex, compact polyhedron  with non-empty interior, given by (\ref{polyhedron}).
By \cite{Schw0},  there exist positive numbers $c_i\in \mathbb R$ such that $\displaystyle\sum_{i=1}^m c_iL_i(X)=1$. Replacing each  $L_i$ by $c_iL_i$ we  may assume that
\begin{equation}\label{quanhe-lambda} \sum_{i=1}^m L_i(X)=1.
\end{equation}
Moreover, it is easy to check that for each  $i=1,\cdots,n$,  there exist real numbers $b_{ij}\in \mathbb R, j=1,\cdots,m$ such that
$$ X_i = \sum_{j=1}^m b_{ij}L_j(X). $$
Let us consider the $n\times m$ matrix $\bold{B}:=(b_{ij})_{i=1,\cdots,n; j=1,\cdots,m}$. Then for $X=(X_1,\cdots,X_n)$ and  $L=(L_1,\cdots,L_m)$, we have
$ X^T=\bold{B}\cdot L^T. $
In other words, we have
\begin{equation}\label{eq:matran}X=L\cdot \bold{B}^T.
\end{equation}
Denote $\mathbb R[Y]:=\mathbb R[Y_1,\cdots,Y_m]$, and consider the ring homomorphism
$$\varphi: \mathbb R[Y] \rightarrow \mathbb R[X], \quad Y_i\longmapsto L_i(X), \forall i=1,\cdots,m.$$
It follows from (\ref{quanhe-lambda}) that $\sum_{i=1}^m Y_i-1\in \mbox{Ker}(\varphi)$. Hence we may assume that the ideal $I:=\mbox{Ker}(\varphi)$ is generated by polynomials $R_1(Y),\cdots, R_s(Y)\in \mathbb R[Y]$,
 $$I:=\mbox{Ker}(\varphi)=\seq{R_1(Y),\cdots,R_s(Y)},$$
 where $\sum_{i=1}^m Y_i-1$  is one of the $R_i$'s.

 Note that the homomorphism $\varphi$ induces a ring homomorphism
$$ M_\varphi: \mathcal{M}_t(\mathbb R[Y]) \longrightarrow \mathcal{M}_t(\mathbb R[X]),\quad  \bold{G}=(g_{ij}(Y))\longmapsto (\varphi(g_{ij}(Y))).   $$

\begin{lemma}\label{lm:surjective} The homomorphism $M_\varphi$ is surjective, and
$$ \mathcal{I}:=\mbox{Ker}(M_\varphi)=\seq{R_1(Y)\bold{I}_t,\cdots, R_s(Y)\bold{I}_t},$$
where $\bold{I}_t$ denotes the identity matrix in $\mathcal{M}_t(\mathbb R[Y])$.
\end{lemma}
\begin{proof}  For each $g(X)=\sum_{|\alpha|\leq d}a_\alpha X^\alpha\in \mathbb R[X]$, denote
\begin{equation}\label{pt:tilde}
\widetilde{g}(Y):=\sum_{|\alpha|\leq d}a_\alpha (Y\cdot B^T)^\alpha \big(\sum_{i=1}^mY_i\big)^{d-|\alpha|}\in \mathbb R[Y].
\end{equation}
It is clear that $\widetilde{g}$ is homogeneous of degree $d$. Moreover  $\varphi(\widetilde{g}(Y))=g(X)$. Hence $\varphi$ is surjective.  Then the surjectivity of $M_\varphi$ follows from that of $\varphi$.

On the other hand, $\bold{G}=(g_{ij}(Y))\in \mbox{Ker}(M_\varphi)$ if and only if $g_{ij}\in \mbox{Ker}(\varphi)$ for all $i,j=1,\cdots,t$. Hence for each $i,j=1,\cdots,t$ we have
 $$g_{ij}(Y)=\sum_{k=1}^s a_{ijk}(Y)R_k(Y), \mbox{ for some } a_{ijk}(Y) \in \mathbb R[Y]. $$
Then $\bold{G}$ can be written as
$$\bold{G}=\sum_{k=1}^s R_k\bold{A_k}=\sum_{k=1}^s (R_k\bold{I}_t)\bold{A_k}, $$
where  $\bold{A_k}=(a_{ijk}(Y))\in \mathcal{M}_t(\mathbb R[Y])$ for each $k=1,\cdots, s$. It is equivalent to the fact that $\bold{G}\in \seq{R_1\bold{I}_t,\cdots, R_s\bold{I}_t}.$
The proof is complete.
\end{proof}

 Let $\bold{F}=(f_{ij})\in \mathcal{S}_t(\mathbb R[X])$ be a polynomial matrix of degree $d>0$. Denote $\bold{\widetilde{F}}:=(\widetilde{f_{ij}})\in \mathcal{S}_t(\mathbb R[Y])$, where each $\widetilde{f_{ij}}$ is defined by (\ref{pt:tilde}), which is a homogeneous polynomial of degree $d$.

Assume  $\lambda(\bold{F})$ is an eigenvalue function of $\bold{F}$. It follows from \cite[Theorem 1]{Z} that $\lambda(\bold{F})$  is a continuous function on $f_{ij}(X)$, $i,j=1,\cdots,t$. That is, there exists a continuous funtion $\Lambda : \mathbb R^{t\times t} \rightarrow \mathbb R$ such that $\lambda(\bold{F})=\Lambda(f_{ij}(X))$.  Denote
$\widetilde{\lambda(\bold{F})}(Y):=\Lambda(\widetilde{f_{ij}}(Y))$, which is actually an eigenvalue function of the polynomial matrix $\bold{\widetilde{F}}$.

 Denote $R(Y):=\displaystyle \sum_{i=1}^s R_i^2(Y)$. With the notations given above, we have the following useful lemma.

\begin{lemma}\label{lm:minors} Let $\bold{F}=(f_{ij})\in \mathcal{S}_t(\mathbb R[X])$ be a polynomial matrix of degree $d>0$.  Let $\lambda(\bold{F})$ is an eigenvalue function of $\bold{F}$.    If $\lambda(\bold{F})> 0$ on $P$, then there exists a sufficiently large constant $c$ such that $\widetilde{\lambda(\bold{F})}+cR>0$ on the standard $m$-simplex $ \Delta_m$.  More explicitly, this holds for $c>-m_1/m_2$, where $m_1$ is the minimum of $\widetilde{\lambda(\bold{F})}$ on $\Delta_m$ and $m_2$ is the minimum of the polynomial $R$ on the compact set
$\Delta_m\cap \{y\in \mathbb R^m|\widetilde{\lambda(\bold{F})}(y)\leq 0\}$.
\end{lemma}
\begin{proof}
The proof goes along the same lines as the proof of \cite[Lemma 4]{PR}, using continuity of the function $\widetilde{\lambda(\bold{F})}$.
\end{proof}

Applying this lemma, we have

\begin{lemma}\label{lm:positive-on-simplex} Let $\bold{F}=(f_{ij})\in \mathcal{S}_t(\mathbb R[X])$ be a polynomial matrix of degree $d>0$. Denote
$\bold{\widetilde{F}}:=(\widetilde{f_{ij}})\in \mathcal{S}_t(\mathbb R[Y])$.
 Assume $\bold{F}\succ \bold{0}$ on $ P$. Then there exists a sufficiently large constant  $c$ such that $\widetilde{\bold{F}}+cR\bold{I}_t \succ \bold{0}$ on the standard $m$-simplex  $\Delta_m$.
\end{lemma}

\begin{proof}  Since $\bold{F}$ is positive definite on $P$, its eigenvalue functions
$\lambda_k(\bold{F}), k=1,\cdots,t$, are positive   on $P$. It follows from Lemma \ref{lm:minors} that for each $k$, there exists a sufficiently large constant $c_k$ such that $\widetilde{\lambda_k(\bold{F})} + c_k R$ is positive on $\Delta_m$. Let $ c:=\displaystyle\max_{k=1,\cdots,t}{c_k}$. Then $\widetilde{\lambda_k(\bold{F})} + c R$ is positive on $\Delta_m$ for each $k=1, \cdots, t$. Note that, $\widetilde{\lambda_k(\bold{F})}$, $k=1,\cdots,t$, are eigenvalues of the polynomial matrix $\widetilde{\bold{F}}$. Moreover, the eigenvalues of the matrix $\widetilde{\bold{F}}+cR\bold{I}_t$ are $\widetilde{\lambda_k(\bold{F})}+cR$, $k=1,\cdots,t$. It follows that  $\widetilde{\bold{F}}+cr\bold{I}_t$  is positive definite on $\Delta_m$. The proof is complete.
\end{proof}

Note that $\overline{\bold{F}}:=\widetilde{\bold{F}}+cR\bold{I}_t$ need not be homogeneous. However, by homogenization $\overline{\bold{F}}$ by $\displaystyle\sum_{i=1}^m Y_i$, we obtain a homogeneous polynomial matrix of the same degree as $\overline{\bold{F}}$. More explicitly, if we express $\overline{\bold{F}}$ as
$$\overline{\bold{F}} = \sum_{|\beta|\leq d} \bold{\overline{F}}_\beta Y^\beta,\quad \bold{\overline{F}}_\beta\in \mathcal{S}_t(\mathbb R), $$
then its homogenization by $\displaystyle\sum_{i=1}^m Y_i$ is
\begin{equation}\label{pt:homogenization} \overline{\bold{F}}^h=\sum_{|\beta|\leq d} \bold{\overline{F}}_\beta Y^\beta (\sum_{i=1}^m Y_i)^{d-|\beta|}.
\end{equation}
  $ \overline{\bold{F}}^h$ is a homogeneous   polynomial matrix of degree $d$. Moreover,
$M_\varphi(\overline{\bold{F}}^h)=\bold{F}$, and $\overline{\bold{F}}^h$ is positive definite on $\Delta_m$.

 Now we can state and prove the following matrix version of Handelman's Positivstellensatz.

\begin{theorem}\label{thr:main} Let $P$, $\bold{F}$, $\overline{\bold{F}}$, $\overline{\bold{F}}^h$ be given as above, with $\bold{F}$  positive definite on $P$. Assume that  $\overline{\bold{F}}^h\succcurlyeq \lambda \bold{I}_t$ on $\Delta_m$ for some $\lambda >0$.  Let  $d:=\mbox{deg}(\overline{\bold{F}})$ and $C:=C(\overline{\bold{F}}^h)$.  Then for each $N> \dfrac{d(d-1)}{2} \dfrac{C}{\lambda}-d$, $\bold{F}$ can be represented as
\begin{equation}\label{equ-representation}
\bold{F}=\sum_{|\alpha|=N+d}\bold{F}_\alpha L_1^{\alpha_1}\cdots L_m^{\alpha_m},
\end{equation}
where  each  $\bold{F}_\alpha\in \mathcal{S}_t(\mathbb R)$ is positive  definite.
\end{theorem}
\begin{proof}
Firstly, applying  the matrix version of P\'{o}lya's Positivstellensatz given in  \cite[Theorem 3]{SchH} for $\overline{\bold{F}}^h$, observing that $d=\deg(\overline{\bold{F}}^h)$. Then, applying $M_\varphi$, using the fact that $M_\varphi(\overline{\bold{F}}^h)=\bold{F}$ and $\varphi(\displaystyle\sum_{i=1}^m Y_i)=1$.
\end{proof}

As a summary, we formulate the construction given  above  as a procedure  to find a   representation
for the  polynomial matrix $\bold{F}=(f_{ij})\in \mathcal{S}_t(\mathbb R[X])$ positive definite on  a convex, compact polyhedron $P\subseteq \mathbb R^n$ as follows:

\begin{itemize}
\item[(1)] Following \cite{Ha} to find positive constants $c_i\in \mathbb R$ such that $\sum_{i=1}^m c_iL_i(X) = 1$. Constructing the $c_i$'s comes down to find a positive solution to an under-determined linear system.
\item[(2)] Solving the system of equations
$$X_i=\sum_{j=1}^m b_{ij}L_{i}(X), \quad i=1,\cdots,n, $$
to find the matrix $\bold{B}=(b_{ij})_{i=1,\cdots,n; j=1,\cdots,m}$.
\item[(3)] Using (\ref{pt:tilde}) to find $\widetilde{f_{ij}}$, $i,j=1,\cdots,t$.
\item[(4)] Using  Gr\"{o}bner bases to find a basis $\{R_1,\cdots,R_s\}$ for the kernel $\mbox{Ker}(\varphi)$ of the ring homomorphism $\varphi$.
\item[(5)] Following the proof of Lemma \ref{lm:positive-on-simplex} to find a sufficiently large $c$ such that $\bold{\widetilde{F}}+c R\bold{I}_t \succ \bold{0}$ on $\Delta_m$.
\item[(6)] Using (\ref{pt:homogenization}) to construct the homogenization $\overline{\bold{F}}^h$ of $\overline{\bold{F}}:=\bold{\widetilde{F}}+c  R \bold{I}_t$.
\item[(7)] Following the proof of  Lemma \ref{lm:compact} below to find the positive number $\lambda$ such that $\overline{\bold{F}}^h(y)\succcurlyeq \lambda \bold{I}_t$ for all $y\in \Delta_m$.
\end{itemize}
\begin{lemma} \label{lm:compact}Let $K\subseteq \mathbb R^m$ be a non-empty compact set, and $\bold{G}\in \mathcal{S}_t(\mathbb R[Y])$. Then there exists a number $c \in \mathbb R$ such that
$$\bold{G}(y)\succcurlyeq   c\bold{I}_t, \mbox{ for all } y \in K.  $$
In particular, if $\bold{G}(y)\succ 0$ for all $y\in K$, then we can choose a number  $c >0 $ such that  $\bold{G}(y)\succcurlyeq  c \bold{I}_t, \mbox{ for all } y \in K.  $
\end{lemma}
\begin{proof} Let $\lambda_{1}(\bold{G}), \cdots, \lambda_t(\bold{G})$ be (real-valued) eigenvalue functions of the polynomial matrix $\bold{G}\in \mathcal{S}_t(\mathbb R[Y]). $ It follows from \cite[Theorem 1]{Z} that $\lambda_i(\bold{G})$ are continuous functions. Since $K$ is compact, let
$$c_i:= \min_{y\in K} \lambda_i(\bold{G})(y), \quad i=1,\cdots, t.$$
Denote $c:=\min_{i=1,\cdots,t} c_i$. Since eigenvalue functions of $\bold{G}-c\bold{I}_t$ are $\lambda_i(\bold{G})-c$, $i=1,\cdots,t$, it follows from the definition of $c$ that
$$ \lambda_i(\bold{G})(y) - c \geq \lambda_i(\bold{G})(y) -c_i\geq 0$$
for all $y\in K$ and for all $i=1,\cdots,t$. This implies that
$\bold{G}(y)\succcurlyeq   c\bold{I}_t, \mbox{ for all } y \in K.$
\end{proof}
\begin{itemize}
\item[(8)] Using the formula (\ref{pt:norm}) to find the number $C:=C(\overline{\bold{F}}^h)$.
\item[(9)] Find a number $N> \dfrac{d(d-1)}{2} \dfrac{C}{\lambda}-d$.
\item[(10)] Find the matrix coefficients of the polynomial matrix $(\sum_{i=1}^m Y_i)^N \overline{\bold{F}}^h\in \mathcal{S}_t(\mathbb R[Y])$, substituting $Y_i$ by $L_i(X)$, we obtain the desired representation for $\bold{F}$.
\end{itemize}

We illustrate the procedure given above by the following example which is computed explicitly using MATLAB Version 7.10 (Release 2010a) and its add-on GloptiPoly 3 discovered by Henrion, Lasserre and L\"ofberg \cite{HLL}.

\begin{example}\rm  Let us consider the unit square centered at the origin
$$P:=\{(x,y)\in \mathbb R^2| L'_1=1+x\geq 0, L'_2=1-x\geq 0, L'_3=1+y\geq 0, L'_4=1-y\geq 0\}. $$
Choosing $c_1=c_2=c_3=c_4=\dfrac{1}{4}$, we have $\sum_{i=1}^4c_iL'_i(x,y)=1$. Therefore, consider
$$L_1:=\dfrac{1}{4}+\dfrac{1}{4}x, ~L_2:=\dfrac{1}{4}-\dfrac{1}{4}x,~ L_3:=\dfrac{1}{4}+\dfrac{1}{4}y, ~L_4:=\dfrac{1}{4}-\dfrac{1}{4}y \in \mathbb R[x,y],$$
 we have $ \sum_{i=1}^4 L_i = 1$.

 It is easy to see that the matrix $\bold{B}=\left[\begin{array}{cccc}
 2&-2&0&0\\
 0&0&2&-2
 \end{array}\right] $ satisfies the equation
 $$\bold{B}\cdot [L_1 ~ L_2 ~ L_3 ~ L_4]^T = [x ~ y]^T.$$

Let $\varphi: \mathbb R[y_1,y_2,y_3,y_4] \rightarrow \mathbb R[x,y]$ be the ring homomorphism defined by $\varphi(y_i):=L_i(x,y)$, $i=1,2,3,4$. Using any monomial ordering in $\mathbb R[y_1,y_2,y_3,y_4]$ we can find a Gr\"{o}bner basis for the kernel $\mbox{Ker}(\varphi)$ of $\varphi$:
$$\{R_1, R_2\}:=\{y_1+y_2-\frac{1}{2}, y_3+y_4-\frac{1}{2}\}.$$
Consider $R:=R_1^2+R_2^2$.

Now we  consider the polynomial matrix $$\bold{F}:=\begin{bmatrix}
-4x^2y+7x^2+y+3&x^3+5xy-3x\\
x^3+5xy-3x&x^4+x^2y+3x^2-4y+6\\
\end{bmatrix}.$$
Eigenvalue functions of $\bold{F}$ are
$$ \lambda_1(\bold{F})=6x^2 - 4x^2y - 4y + 6; ~  \lambda_2(\bold{F})=x^4 + x^2y + 4x^2 +y+ 3.$$
For any $(x,y)\in P$ we have $\lambda_i(\bold{F})(x,y)\geq 2$, $i=1,2$. Hence $\bold{F}(x,y)\succcurlyeq 2\bold{I}_2$ for every $(x,y)\in P$.\\
With the matrix $\bold{B}$ considered above, using the formula (\ref{pt:tilde}), we   find   $\widetilde{f_{ij}}$, $i,j=1,2$, and then we obtain  the polynomial matrix  $\widetilde{\bold{F}}=(\widetilde{f}_{ij})$. We can compute exactly the eigenvalue functions $\lambda_1(\widetilde{\bold{F}})$ and $\lambda_2(\widetilde{\bold{F}})$ of $\widetilde{\bold{F}}$ which satisfy
 $$\min_{\Delta_4}\lambda_1(\widetilde{\bold{F}}) =1,~ \min_{\Delta_4}\lambda_2(\widetilde{\bold{F}}) =-2.$$
Moreover, $\displaystyle \min_{\Delta_4\cap \{\lambda_2(\widetilde{\bold{F}})\leq 0\}} R(y_1,y_2,y_3,y_4) = 0.125$.  Thus we can choose
$$c>-\dfrac{-2}{0.125}=16, \mbox{ namely}, ~c=17,$$
 for which $\overline{\bold{F}}:=\widetilde{\bold{F}}+cR \bold{I}_2 \succ \bold{0} $ on $\Delta_4$.\\
Homogenizing  $\overline{\bold{F}}$ by $\sum\limits_{i=1}^4y_i$ we obtain a homogeneous polynomial matrix  $\overline{\bold{F}}^h=(\overline{f_{ij}}^h)$. Then we compute exactly the eigenvalue functions of the matrix $\overline{\bold{F}}^h$ which satisfy
$$\min_{\Delta_4}\lambda_1(\overline{\bold{F}}^h) = 1.9706,~ \min_{\Delta_4}\lambda_2(\overline{\bold{F}}^h) = 1.5294.$$
It follows that $\overline{\bold{F}}^h\succcurlyeq  1.5294 ~  \bold{ I}_2$ on $\Delta_4$, and put $\lambda:=1.5294$.\\
Using the formula (\ref{pt:norm}), we can find the number $C:=C(\overline{\bold{F}}^h)=\dfrac{1044}{24}=\dfrac{87}{2}$. \\
Therefore,  choosing $N=167$, the polynomial matrix $(y_1+y_2+y_3+y_4)^{167}\overline{\bold{F}}^h$ has positive definite coefficients.\\
Find the matrix coefficients of the polynomial matrix $(y_1+y_2+y_3+y_4)^{167}\overline{\bold{F}}^h\in \mathcal{S}_t(\mathbb R[y_1,y_2,y_3,y_4])$, substituting $y_i$ by $L_i(x,y)$, we obtain the desired representation for $\bold{F}$.
\end{example}

As a consequence of Theorem \ref{thr:main}, we obtain the following matrix version  of Schm\"udgen's Positivstellensatz for convex, compact polyhedra.
\begin{corollary}
Let $P$,  $\bold{F}$, $\overline{\bold{F}}$, $\overline{\bold{F}}^h$ be given as above, with $\bold{F}$  positive definite on $P$. Assume that  $\overline{\bold{F}}^h\succcurlyeq \lambda \bold{I}_t$ on $\Delta_m$ for some $\lambda >0$. Let  $d:=\mbox{deg}(\overline{\bold{F}})$ and $C:=C(\overline{\bold{F}}^h)$.  Then for $N> \dfrac{d(d-1)}{2} \dfrac{C}{\lambda}-d$, $\bold{F}$ can be represented as
\begin{equation}\label{coro-equ-representation}
\bold{F}=\sum_{e=(e_1,\cdots,e_m)\in \{0,1\}^m}(\bold{F}_e^T \bold{F}_e) L_1^{e_1}\cdots L_m^{e_m},
\end{equation}
where   $\bold{F}_e\in \mathcal{M}_t(\mathbb R[X])$  and the degree of each sum of squares  $\bold{F}_e^T\bold{F}_e$ does not exceed $N+d$.
\end{corollary}
\begin{proof} The proof follows directly from Theorem \ref{thr:main}, with the observation that any positive definite matrix $\bold{F_\alpha}\in \mathcal{S}_t(\mathbb R)$ can be written as
$$ \bold{F_\alpha} = \bold{G_\alpha}^T \bold{G_\alpha}, $$
where  $ \bold{G_\alpha}\in \mathcal{M}_t(\mathbb R)$ is a non-singular matrix.

\end{proof}
\textbf{Acknowledgements} ~  The authors would like to thank the anonymous referees for their useful comments and suggestions. They would also like to   thank  Dr. Ngo Lam Xuan Chau for his fruitful discussion to compute the kernel of the homomorphism $M_\varphi$.  Both  authors are partially supported by  the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under   grant number  101.01-2016.27.


\section{Introduction}
GK Persei, hereafter GK Per, is a cataclysmic variable (CV) system that
hosts a magnetized white dwarf (WD) and is observable as an intermediate polar.
This source underwent a classical nova explosion in 1901 \citep{williams1901,hale1901} and also exhibits dwarf nova outbursts about every 3~years each lasting
about 50 days (e.g. \citealt{simon2002}). Although GK Per is a magnetic
system that has magnetically channeled accretion within its magnetosphere,
it is believed that the dwarf nova outburst occurs due to increased
mass transfer caused by instability within an accretion disk circulating
around the magnetosphere of the WD (e.g. \citealt{nogamietal2002}).
\citet{morales-ruedaetal2002gkper} measured an absorption spectrum of the companion
star, and reported mass ratio of the companion and the WD as $q=M_\mathrm{K}/M_\mathrm{WD}=0.55\pm0.21$, and constrained the WD mass as $M_\mathrm{WD}>0.87\pm0.24~M_\odot$. \citet{suleimanovetal2005} estimated a WD mass
based on X-ray spectral fitting as $M_\mathrm{WD}=0.59\pm0.05~M_\odot$ noting that the value may be underestimated by $\sim20\%$ due to reduced magnetospheric radius in outburst in which their {\it RXTE}  data were taken. \citet{hachisukato2007gkper} derived the best-fit WD mass of $1.15\pm0.1~M_\odot$ based on the classical nova light curve modeling.

Because of its unique characteristics (i.e. magnetic CV with classical nova/dwarf nova activies), GK Per has been deeply studied in broad
wavelength; see \citealt{nogamietal2002} and \citealt{takeietal2015} for extensive
collections of previous publications.
Among many studies, key milestones in X-ray, that are relevant to the present paper,
include detections of
(1) the spin period of 351~s using {\it EXOSAT} \citep{watsonetal1985},
(2) the complex absorption structure and Fe K$\alpha$ emission line structure using {\it Ginga}  and {\it ASCA}  \citep{ishidaetal1992, ezukaishida1999},
(3) the potential red-ward wing in Fe K$\alpha$ line using {\it Chandra}  HETG \citep{hellierandmukai2004fek}, and
(4) emission lines from lighter elements such as O and N using {\it XMM-Newton}  RGS \citep{vrielmannetal2005}.
Studies of the X-ray emitting expanding shell which was produced by the 1901 nova suggested presence of non-equilibrium ionization collisional plasma in the nova remnant \citep{balman2005,takeietal2015}.

Dwarf nova outbursts of GK Per provide us with interesting opportunities to study
disk instability \citep{kimetal1992,nogamietal2002} and a disk-magnetosphere
interaction in a changing mass accretion rate ($\sim\times10-20$ compared to
that of quiescence; \citealt{ishidaetal1992}).
In the present paper, we analyze X-ray observation data of the latest dwarf nova
occurred in March-April 2015 to investigate the accretion geometry and the structure
of the post-shock accretion region such as the maximum temperature and
the magnetospheric radius in outburst.
We also compare results of these analyses with those obtained in quiescent when available.

\section[]{Observation and data reduction}\label{sec:obs}

On March 6.84 UTC of 2015 (MJD 57087.84), GK Per was found to have transited into
the dwarf nova outburst state \citep{wilberetal2015}.
{\it Suzaku}  X-ray satellite was coincidently observing GK Per as a scheduled
observation from 2015-03-05 18:38:56 UTC (MJD 57086.8)
to 2015-03-09 18:30:11 UTC (MJD 57090.8)
with net exposures of 182.9~ks with the X-ray Imaging Spectrometer
(XIS; \citealt{koyamaetal2007xis}) and 134.8~ks with the Hard X-ray Detector
(HXD; \citealt{takahashietal2007,kokubunetal2007}).
Table \ref{tab:obslog} summarizes the observation log.
Note that due to the degraded power generation capability of the spacecraft,
only one XIS unit among four, namely XIS3, was operational during this observation.
In addition, the HXD was operational while the solar paddles were illuminated by the sun, and turned off in the eclipse by the earth.
As shown in Figure \ref{fig:swiftLC}, the {\it Suzaku}  observation covered
a transition from quiescent to outburst, allowing us to trace temporal variation of X-ray emission;
the figure utilizes Swift/BAT transient monitor results provided by the Swift/BAT team \citep{krimmetal2013}.

Besides the March 2015 observation, {\it Suzaku}  has been pointed to the
target twice in February 2009 and August 2014 when the target was in
the quiescent state in optical (Table \ref{tab:obslog}).
Data from these observations are also analyzed in the present report.
We remark the following two points on these observations;
(1) although three XIS units and the HXD were operational in February 2009,
the source flux was too low to be detected by the HXD,
(2) in August 2014, three XIS units were operational but the HXD
was accidentally left turned off by the cancellation of spacecraft contact operations
due to typhoon interference.

The March 2015 data, together with those from the 2009 and 2014 observations,
were processed and analyzed using HEASOFT 6.15 maintained and released
by the HEASARC at NASA Goddard Space Flight Center.
We followed the standard data extraction and event selection procedures of {\it Suzaku}.
The background signals of the XIS was estimated and subtracted using data
detected in the outer region of the imaging area around the source image.
The HXD/PIN non-X-ray background was subtracted
using the official background model \citep{fukazawaetal2009} released by the {\it Suzaku}  HXD team
\footnote{https://heasarc.gsfc.nasa.gov/docs/suzaku/analysis/pinbgd.html}.
The barycentric correction was applied to arrival time of each photon recorded by the XIS and the HXD.

\begin{table*}
\begin{center}
\begin{minipage}{140mm}
\caption{Log of GK Per observations by {\it Suzaku}.}
\label{tab:obslog}
\begin{tabular}{cccccccc}
\hline
Observation ID & Start Date/Time & Exposure$^{a}$ & Aim point & \# of XIS$^{b}$ & Count rate$^{c}$\\
\hline
403081010 & 2009-02-13T10:35:48 & 30.4 & HXD & 3 & $0.25$ \\
409018010 & 2014-08-07T17:01:21 & 114.5 & XIS & 3 & $0.89$ \\
409018020 & 2015-03-05T18:38:56 & 182.9 & XIS & 1 & $3.08$ \\
\hline
\end{tabular}\\
$^{a}$Effective exposure of the XIS in ks.
$^{b}$The number of the XIS unit(s) operational in the observation.
$^{c}$Average $0.5-10$~keV count rates recorded by the XIS FI sensors
in units of counts~s$^{-1}$~XIS$^{-1}$. Statistical errors are less than 0.01 counts~s$^{-1}$.
Note that in the HXD-nominal aim point observation, where a target
point-like source is offset by 3.5~arcmin from the XIS-nominal aim point,
results in $\sim15-20\%$ smaller effective area due to the energy-dependent
vignetting effect of the X-ray mirror.
\end{minipage}
\end{center}
\end{table*}

\begin{figure*}
\includegraphics[width=0.85\hsize]{figures/swiftLC.pdf}
\caption{
Light curve of {\it Swift}/BAT $15-50$~keV count rate obtained from
the {\it Swift}/BAT Hard X-ray Transient Monitor website.
{\it Suzaku} observation period is shown as shaded region.}
\label{fig:swiftLC}
\end{figure*}

\section{Analysis and results}\label{sec:analysis}
\subsection{Temporal analysis}\label{sec:temporalAnalysis}

\subsubsection{Light curves}\label{sec:lc}
Figure \ref{fig:xisLC} presents light curves of count-rate and hardness ratio.
Apparently, the March 2015 observation nicely covered the quiescent state
(from 0 to around $5\times10^{4}$~s) and the onset of the dwarf nova outburst.
The count rates of the quiescent period of the March 2015 observation
are almost the same as that of August 2014;
$\sim0.1-0.3$~counts~s$^{-1}$ ($0.5-2$~keV) and
$\sim1$~counts~s$^{-1}$ ($2-10$~keV).
In the later period of the March 2015 observation, the count rates are rather stable at $\sim0.1-0.2$~counts~s$^{-1}$ ($0.5-2$~keV)
and $\sim3.5-4.5$~counts~s$^{-1}$ ($2-10$~keV).

Hard X-ray ($>16$~keV) light curves of the March 2015 observation are plotted
in Figure \ref{fig:hxd_lightcurve}.
In contrast to the soft-band signals below 10~keV (Figure \ref{fig:xisLC} bottom three panels),
the hardness ratio calculated in the HXD/PIN energy band is more stable,
suggesting less change in spectral shape than in the softer energy band.

In the following analyses, to study temporal evolution of spin profiles and
spectral shapes, we divided the March 2015 data into six equally separated
57320-s time intervals ($\sim0.68$~day) each having an effective XIS exposure of
$\sim30$~ks, and labeled as Epoch 0 to Epoch 5.
The start time of Epoch 0 was set at the time of the first good time interval of the XIS (2015-03-05 18:55:48.7 UTC; MJD 57086.78954).
During the 2009 and the 2014 observations, GK Per did not
show significant time variability, and therefore, we analyzed their data
without splitting into shorter time epochs to produce statistically significant data sets.

\begin{figure*}
\includegraphics[width=0.85\hsize]{figures/fi_evt_source_ra_dec_120_1st.pdf}
\includegraphics[width=0.85\hsize]{figures/fi_evt_source_ra_dec_120_2nd.pdf}
\includegraphics[width=0.85\hsize]{figures/fi_evt_source_ra_dec_120_3rd.pdf}
\caption{
Light curves of XIS count rates of the three observations in February 2009, August 2014, and March 2015. The top, the middle, and the bottom panels in each plot shows a soft-band count rate ($0.5-2$~keV), a hard-band count rate ($2-10$~keV), and the hardness ratio calculated as ($2-10$~keV count rate)/($0.5-2$~keV count rate). Count rate is scaled per XIS unit. Horizontal lines with number labels in the middle panel of the bottom plot represent epochs defined for analyses of the March 2015 data (see text).
}
\label{fig:xisLC}
\end{figure*}

\begin{figure}
\includegraphics[width=1\hsize]{figures/hxd_lightcurve.pdf}
\caption{
HXD/PIN count-rate light curves in the $16-25$~keV band (top panel) and the $25-60$~keV band (middle panel) of the March 2015 observation. Hardness ratio, calculated as a ratio of count rates ($16-25$~keV)/($25-60$~keV), is plotted in the bottom panel.
}
\label{fig:hxd_lightcurve}
\end{figure}

\begin{figure*}
\includegraphics[height=0.68\hsize]{figures/folded_lc_soft.pdf}
\includegraphics[height=0.68\hsize]{figures/folded_lc_hard.pdf}
\includegraphics[height=0.68\hsize]{figures/folded_lc_all_16_60.pdf}
\caption{
Time evolution of spin-folded light curves of the March 2015 data in three energy bands;
$0.5-2$, $2-10$, and $16-60$~keV in left, middle, right columns, respectively.
From the top to the bottom, the six panels in each column represent light curves for Epoch 0 to 5.
In each panel, the abscissa covers two spin phases for better visibility.
}
\label{fig:spin_profiles}
\end{figure*}

\subsubsection{Spin profiles}\label{sec:lc}
Using the March 2015 XIS data, we generated
a power spectrum and a periodgram, and obtained a WD spin period of
$P_{\mathrm{WD}}=351.4\pm0.5$~s.
Based on this, energy-sliced spin-folded light curves were extracted from
the XIS and the HXD/PIN data for each time epoch as shown in Figure \ref{fig:spin_profiles}.
In the folding process, the phase 0 epoch was set at 2015-03-05 00:00:00 UTC (MJD 57086.0), and a spin period of 351.4~s was assumed.
Table \ref{tab:xis_folded_lc} summarizes the mean count rate and the modulation factor
calculated as the full amplitude divided by the mean count rate.

In the figure, a complex time evolution of the spin profiles is seen; for example,
the spin minimum phase shifts from phase $0.2-0.3$ in Epochs $0-2$ to $0.6-0.7$
in Epochs $4-5$, and the pulse shape significantly changes epoch by epoch.
As previously reported for the outburst state \citep{watsonetal1985,nortonetal1988,evansetal2009},
the overall pulse profile becomes single peaked, especially in the later epochs.
Throughout the epochs, the modulation factors are higher in the soft band than
those in the hard band.
The hard X-ray folded light curve also modulates with a modulation factor of $\sim40\%$
(or a min/max ratio of $\sim0.7$) in outburst.

\begin{table*}
\begin{center}
\begin{minipage}{160mm}
\caption{Mean count rate and modulation factor of the spin-folded light curves of the March 2015 outburst. Mean count rates of the 2009 and the 2014 data are also listed for comparison.}
\label{tab:xis_folded_lc}
\begin{tabular}{cccccccccc}
\hline
 & \multicolumn{2}{c}{$0.5-2$ keV} &  & \multicolumn{2}{c}{$2-10$ keV} &  & \multicolumn{2}{c}{$16-60$ keV}\\ \cline{2-3} \cline{5-6} \cline{8-9}
         & Mean$^{a}$ & MF$^{b}$ & & Mean$^{a}$ & MF$^{b}$ & & Mean$^{a}$ & MF$^{b}$\\
\hline
2009 & $0.104\pm0.002$ & $-$ & & $0.151\pm0.002$ & $-$ & & N/A$^{c}$ & $-$ \\
2014 & $0.162\pm0.001$ & $-$ & & $0.730\pm0.002$ & $-$ & & N/A$^{c}$ & $-$ \\
2015\\
\input{tables/folded_lc_table.tex}
\hline
\end{tabular}
\\
$^{a}$In units of counts~s$^{-1} \mathrm{XIS}^{-1}$.
$^{b}$Modulation factor in \%. See text for definition.\\
$^{c}$GK Per flux was below the HXD sensitivity, or the HXD was not operational (see \S\ref{sec:obs}).
\end{minipage}
\end{center}
\end{table*}

\subsection{Spectral analysis}\label{sec:spectral}

\subsubsection{Spectral evolution in the soft energy band}\label{sec:spectral_xis}
Figure \ref{fig:xis_123} compares time-average spectra of the three observations.
Obviously, the source is more luminous in outburst than in quiescence showing about 2 orders of magnitude increase from February 2009 in $>7$~keV. Among the quiescent states, however, detected fluxes differ by a factor of $>5$ (see $2-10$~keV fluxes in Table \ref{tab:xis_nH}), implying a temporal fluctuation of mass accretion rate even during the quiescence. Based on the intensity-scaled spectra in the right of Figure \ref{fig:xis_123}, a major spectral change is observed below 4~keV, and above Fe K line complex ($>7$~keV), the spectra well overlap each other.

For the outburst data, we also extracted time-resolved spectra for the six epochs as presented in Figure \ref{fig:xis_3rd_spectra_all_intervals}. The shape and the normalization of the black spectrum (Epoch 0) almost matches that of the August 2014 time-average spectrum.
Based on this, we consider that the source was in the quiescent state in Epoch 0, and then a transition to the dwarf nova outburst took place during Epoch 1 where steady increases of the $2-10$~keV and the $16-60$~keV count rates were observed (Figures \ref{fig:xisLC} and \ref{fig:hxd_lightcurve}). The right panel of Figure \ref{fig:xis_3rd_spectra_all_intervals} plots intensity-scaled spectra of individual epochs. As the dwarf nova outburst proceeds, soft X-ray photons below $\sim4$~keV drastically decreased.
This is a firm indication of an increased absorption column density during the outburst.

In the outburst phase, for example in the most absorbed spectrum of Epoch 5, we detected $0.027\pm0.001$~counts~s$^{-1}$ in the $0.5-1$~keV band, and this rate is 2 or 3 times higher than one expected from heavily ($N_\mathrm{H}>10^{22}$~cm$^{-2}$) absorbed post-shock plasma emission (i.e. extrapolation of $E>2$~keV emission to lower energy band).
Based on the previous grating observation results \citep{vrielmannetal2005}, we consider that this emission is mostly a sum of line emissions whose origin has not been clearly identified. A detailed spectral modeling of this emission is very challenging with the limited energy resolution of the XIS; we tried to fit this component with multiple gaussian components or phenomenological continuum represented by black body emission but no acceptable fit was obtained. Therefore, in the following spectral fittings, we ignore energies below 2~keV to avoid a complication caused by this soft emission.

\subsubsection{Absorption column density}\label{sec:xis_nH}
As can be seen from Figures \ref{fig:xis_123} and \ref{fig:xis_3rd_spectra_all_intervals},
absorption of soft X-ray photons is highly variable even in the quiescent state, and obviously during outburst.
The absorption is thought to be mostly intrinsic to GK Per caused by cool pre-shock accreting gas rather than by interstellar medium. To examine an absorption column density increase, we fitted the XIS spectra using a phenomenological model consisting of a bremsstrahlung continuum subject to absorption. The Fe K emission line region, $5.5-7.2$~keV, was ignored to simplify the fitting model. In Epochs $2-5$, a single-absorption model did not reproduce the data, and therefore we introduced a partially-covering absorber.
Fits are all acceptable, and results are summarized in Table \ref{tab:xis_nH}. The overall absorption column density increases by a factor of $\sim2.4$; from $1.82^{+0.15}_{-0.15}\times10^{22}~\mathrm{cm}^{-2}$ (Epoch 0) to the highest $4.31^{+0.42}_{-0.56}\times10^{22}~\mathrm{cm}^{-2}$ (Epoch 4). The $7.2-10$~keV flux increases by a factor of $\sim4.8$; from 1.35 (Epoch 0) to $6.46\times10^{-11}~\mathrm{erg}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}$ (Epoch 5). Higher partial-covering column densities $\sim13-16\times10^{22}~\mathrm{cm}^{-2}$ was obtained in later epochs and we consider that spin modulation of the line-of-sight column density of this dense absorber creates the hard X-ray modulation detected by the HXD (Figure \ref{fig:spin_profiles}).
Although the face values of bremsstrahlung temperature vary from $19.2^{+7.4}_{-4.5}$~keV (Feb 2009) to $64.4^{+20.8}_{-14.5}$~keV (Epoch 1 of Mar 2015), this is not considered as a real variation because temperatures cannot be accurately determined solely from the XIS data when thermal emission has a temperature higher $kT\sim10$~keV (see \S\ref{sec:spectral_hxd}).

\begin{table*}
\begin{center}
\begin{minipage}{14cm}
\caption{Empirical spectral modeling of the $2-10$~keV continuum emission using a bremsstrahlung continuum and three gaussian components.}
\label{tab:xis_nH}
\begin{tabular}{cccccccc}
\hline
	& $N_{\mathrm{H}}$	& $N^{\mathrm{PC}}_{\mathrm{H}}$$^{a}$	& $kT$ & C.F.$^{b}$ & \multicolumn{2}{c}{Flux$^{c}$} & $\chi^2_{\nu}$~(N.D.F.)$^{d}$ \\
	& $10^{22}~\mathrm{cm}^{-2}$	& $10^{22}~\mathrm{cm}^{-2}$	& keV & & $2-10~\mathrm{keV}$ & $7.2-10~\mathrm{keV}$  \\
\hline
2009 & $0.24^{+0.23}_{-0.22}$ & $-$ & $19.2^{+7.4}_{-4.5}$ & $-$ & 0.72 & 0.18 & 1.26(137)\\
2014 & $1.59^{+0.06}_{-0.06}$ & $-$ & $32.0^{+3.0}_{-2.6}$ & $-$ & 3.1 & 0.93 & 1.25(390)\\
2015\\
\input{tables/table_phabs_brems_phabs_brems.tex}
\hline
\end{tabular}\\
$^{a}$Column density of the partial-covering absorption applied to Epoch 2-5 of the 2015 data.\\
$^{b}$Covering fraction of the partial-covering absorber.\\
$^{c}$Model flux in the $2-10$~keV and the $7.2-10$~keV bands in units of $10^{-11}~\mathrm{erg}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}$. The former does not include $5.5-7.2$~keV flux.\\
$^{d}$Reduced $\chi^2$ and the number of degree of freedom.
\end{minipage}
\end{center}
\end{table*}

\subsubsection{Fe K$\alpha$ lines}\label{sec:xis_fe_line}
To examine possible time variability of Fe K$\alpha$ line intensities, we also fitted $4-9$ keV narrow-band spectra using an empirical model consisting of a power-law continuum and three gaussians all subject to single absorption. Centroid energies and widths of the three lines were fixed at canonical values (6.40, 6.65, and 6.97~keV) and 0 (i.e. no broadening), respectively, so as to better constrain equivalent widths. The best-fit model functions are plotted in Figure \ref{fig:fe_line_fit} together with the data. Fits were generally acceptable, and we obtained line equivalent widths as listed in Table \ref{tab:fe_line} and plotted in Figure \ref{fig:fe_line}.

The fluorescent Fe K$\alpha$ line increased its equivalent width from $\sim80$~eV in quiescent to $\sim140$~eV in outburst. The equivalent widths of the He-like and H-like Fe K$\alpha$ lines are in the range of $30-50$~eV, and are consistent with being constant throughout the present data set; this is qualitatively consistent with the fact that the continuum spectral shape, in the $7-10$~keV, are very similar in all data sets (Figure \ref{fig:xis_123} and \ref{fig:xis_3rd_spectra_all_intervals}) indicating little change in plasma temperature.

\begin{figure*}
\includegraphics[width=0.48\hsize]{figures/xis_123.pdf}
\includegraphics[width=0.48\hsize]{figures/xis_123_match7_10keV.pdf}
\caption{
Left: Time-average XIS spectra of the three observations;
green, orange, and dark grey spectra corresponds to observations in February 2009, August 2014, and March 2015, respectively. Ordinate is shown after scaling the number of XIS units operated in each observation.
Right: The same spectra as those in the left panel but vertically scaled for the $7-10$~keV band count rate to match each other; scaling factors are 1, 0.15, and 0.06 for spectra of February 2009, August 2014, and March 2015, respectively.
}
\label{fig:xis_123}
\end{figure*}

\begin{figure*}
\includegraphics[width=0.48\hsize]{figures/xis_3rd_spectra_all_intervals.pdf}
\includegraphics[width=0.48\hsize]{figures/xis_3rd_spectra_all_intervals_scaled.pdf}
\caption{
Left: XIS spectra of individual epochs of the March 2015 observation.
Right: The same spectra as those in the left panel but vertically scaled for the $7-10$~keV band count rate to match each other; scaling factors are 1, 0.48, 0.29, 0.24, 0.23, and 0.23 for Epochs $0-5$, respectively.
}
\label{fig:xis_3rd_spectra_all_intervals}
\end{figure*}

\begin{figure}
\includegraphics[width=1\hsize]{figures/intervals_model_wabs_pow_gaus3.pdf}
\caption{
Fe K line fit with three gaussian components and a power-law continuum (top panel).
For better visibility, fit residuals of Epoch 0-2 and 3-5 are separately plotted in the middle and the bottom panels, respectively. Color coding is the same as that of Figure \ref{fig:xis_3rd_spectra_all_intervals}.
}
\label{fig:fe_line_fit}
\end{figure}

\begin{figure}
\includegraphics[width=0.95\hsize]{figures/fe_line.pdf}
\caption{
Fe K line equivalent widths derived by an empirical model fitting.
Equivalent widths of the fluorescent, He-like, and H-like Fe K$\alpha$
lines are represented in black, red, and green, respectively.
E0$-$E5 denotes Epoch $0-5$ of the March 2015 observation.
}
\label{fig:fe_line}
\end{figure}

\begin{table}
\begin{center}
\caption{Fe K$\alpha$ line equivalent widths.}
\label{tab:fe_line}
\input{tables/table_xis_fe_line.tex}
\end{center}
\end{table}

\subsubsection{Hard X-ray spectra}\label{sec:spectral_hxd}
To characterize the hard-band spectrum during the transition to the outburst state,
we extracted epoch-separated HXD/PIN spectra, and fitted with two phenomenological models;
(1) a power-law model and (2) a thermal bremsstrahlung model.
The fit range was chosen to be $16-60$~keV (the same as the light-curve extraction energy range).
The extracted spectra and the best-fit
model functions are shown in Figure \ref{fig:hxd_fit_brems}, and the best-fit parameters
are listed in Table \ref{table:hxd_fit_pow_brems}.
The best-fit photon indices and temperatures do not show large variation across epochs,
and are almost consistent each other within statistical fitting errors; $\Gamma=2.2^{+0.3}_{-0.3} - 2.4^{+0.1}_{-0.1}$ and $kT=24.6^{+1.5}_{-1.4} - 31.5^{+15.5}_{-8.5}$~keV.
This is another result supporting non-variability of the intrinsic spectral shape of emission from the post-shock accretion region throughout the quiescence and the early phase of the outburst. In Epoch 0, an apparent deviation of data and the best-fit model curve in the $40-60$~keV band is noticeable in Figure \ref{fig:hxd_fit_brems}.
This excess cannot be solely explained by the inaccurate background modeling as one might consider, because even in Epoch 0 (quiescent state), the $40-60$-keV band count rate ($0.01$~counts~s$^-1$) is an order of magnitude higher than that of the background model accuracy in the same energy band. Although we do not fully understand the origin of the excess, exclusion of the energy band from the fitting results similar best-fit parameters as listed in Table \ref{table:hxd_fit_pow_brems}, and therefore we leave the excess as is in the present analysis.

\begin{figure}
\includegraphics[width=1\hsize]{figures/hxd_fit_brems.pdf}
\caption{
Upper panel: Time-divided HXD/PIN spectra of GK Per taken in the third observation.
Crosses are the background-subtracted signals, and the solid lines represent the best-fit bremsstrahlung model functions.
Lower panel: Fit residual in terms of $\chi$ (data$-$model divided by statistical error).
}
\label{fig:hxd_fit_brems}
\end{figure}

\begin{table*}
\begin{center}
\begin{minipage}{140mm}
\caption{HXD/PIN spectral fit result of the March 2015 data using a power-law model or a bremsstrahlung model.}
\label{table:hxd_fit_pow_brems}
\input{tables/table_hxd_pow_brems.tex}
\\
$^{a}$Power-law photon index.
$^{b}$Reduced $\chi^2$ and the number of degree of freedom.\\
$^{c}$Model predicted $16-60$~keV flux in units of $10^{-11}~\mathrm{erg}\ \mathrm{cm}^{-2}\ \mathrm{s}^{-1}$.
$^{d}$Temperature in keV.
\end{minipage}
\end{center}
\end{table*}

\subsubsection{Broad-band spectral interpretation}\label{sec:spectral_broad}
We also fitted broad-band combined spectra of the XIS and the HXD/PIN
in the $2-60$~keV. The broad-band model consists of an isobaric cooling-flow
plasma emission model (developed for galaxy cluster studies; \citealt{mushotzkyzymkowiak1988})
subject to a single-column photo absorption and a partial-covering
photo absorption to mimic multi-column absorption \citep{donemagdziarz1998} and spin modulation of
column density. Reflection on freshly accreted material on the WD surface
is modeled using the convolution model \verb|reflect| by \citealt{magdziarzandzdziarski1995}.
A gaussian is utilized to model the fluorescent Fe K$\alpha$ line emitted
by the reflection. The composite model can be denoted in Xspec as
\verb|phabs|$\times$\verb|pcfabs|$\times($\verb|reflect|$\times$\verb|mkcflow|$+$
\verb|gaus|$)$.
The cooling-flow model is a good approximation to the X-ray emission from
the intermediate polars' post-shock accretion region (\citealt{mukaietal2015,hayashietal2011}), and in the present analysis,
we utilized the model aiming at assessing a potential change of the maximum temperature in the post-shock accretion region.
The reflector's covering area against the irradiator (i.e. hot plasma in the post-shock accretion region) and relative normalization of the reflection are fixed at $2\pi$ and $1$, respectively, assuming that the shock height is much smaller than the radius of the WD (for example, see \citealt{yuasaetal2010}). Since the angle dependent reflection was hardly detected in the present data, we tentatively fixed the reflection angle, or an angle between the reflecting surface normal and the line of sight, at the default value of $\cos\mu=0.45$. Fe abundance of the cooling-flow model and the reflection model are tied and allowed to freely vary.

The spectra of individual epochs were separately fitted, and the best-fit results were obtained as plotted in Figure \ref{fig:broad_band_fit}. Table \ref{tab:broad_band_fit} summarizes the best-fit parameters. The fits are generally good with the null hypothesis probabilities larger than $1\%$ all except for Epoch 5 ($0.1\%$). The overall absorption column density, the first column of Table \ref{tab:broad_band_fit}, are almost consistent with those derived from the XIS-only fitting (Table \ref{tab:xis_nH}) within the statistical fitting errors, and the tendency of increase over time is common between the two modelings. The apparent increase of the covering fraction may suggest more frequent obscuration of the post-shock plasma by a pre-shock cold gas in later phases of the outburst. The face values of the highest temperature of the cooling-flow model increase through Epoch 0 to Epoch 2, and are consistent with being constant (within errors) in Epochs 2-5.

The maximum temperature reported by the cooling-flow model is generally higher than one obtained with the PIN-only bremsstrahlung fitting (Table 5) in each epoch. This is an expected result because the bremsstrahlung model only provides a representative temperature that simulates a spectrum from a post-shock region that has an emissivity gradient, being weighted by lower-temperature component which has higher emissivity (due to increased density) and photon statistics. When the PIN spectra are fitted with a cooling-flow model, the maximum temperatures similar to those from the XIS+PIN fitting (Table \ref{tab:broad_band_fit}) are obtained; e.g. $kT_{\mathrm{max}}=47.2^{+19.6}_{-11.6}$~keV and $49.0^{+3.5}_{-2.8}$~keV for Epoch 0 and 5, respectively.

\begin{table*}
\begin{center}
\begin{minipage}{14cm}
\caption{Result of broad-band spectral fit using an partially-absorbed multi-temperature emission model convolved with a reflection model.}
\label{tab:broad_band_fit}
\begin{tabular}{cccccccc}
\hline
Epoch & $N_{\mathrm{H}}$$^{\mathrm{a}}$   & $N^{\mathrm{PC}}_{\mathrm{H}}$$^{\mathrm{b}}$ & C.F.$^{\mathrm{c}}$ & $kT_{\mathrm{max}}$$^{\mathrm{d}}$ & $Z_{\mathrm{Fe}}$$^{\mathrm{e}}$ & $\chi^2_{\nu}$~(N.D.F.)$^{f}$\\
      & $10^{22}$~cm$^{-2}$ & $10^{22}$~cm$^{-2}$ &      & keV          & $Z_\odot$          & \\
\hline
0 & $1.79^{+0.16}_{-0.09}$ & $62.2^{+31.5}_{-47.9}$ & $0.17^{+0.17}_{-0.15}$ & $41.8^{+9.8}_{-9.8}$ & $0.10^{+0.06}_{-0.05}$ & $0.94$ (1308)\\
1 & $2.60^{+0.13}_{-0.15}$ & $64.3^{+15.9}_{-20.1}$ & $0.30^{+0.09}_{-0.10}$ & $45.1^{+8.6}_{-6.8}$ & $0.09^{+0.04}_{-0.03}$ & $1.04$ (1782)\\
2 & $3.53^{+0.22}_{-0.29}$ & $23.6^{+8.3}_{-6.7}$ & $0.35^{+0.03}_{-0.03}$ & $59.3^{+4.7}_{-5.0}$ & $0.12^{+0.03}_{-0.02}$ & $0.99$ (1980)\\
3 & $3.75^{+0.16}_{-0.17}$ & $40.7^{+7.4}_{-6.8}$ & $0.45^{+0.03}_{-0.03}$ & $55.2^{+5.1}_{-5.0}$ & $0.08^{+0.02}_{-0.02}$ & $1.06$ (2007)\\
4 & $5.50^{+0.22}_{-0.23}$ & $41.2^{+7.6}_{-6.8}$ & $0.49^{+0.03}_{-0.02}$ & $59.4^{+5.9}_{-5.7}$ & $0.10^{+0.02}_{-0.02}$ & $1.03$ (2011)\\
5 & $4.88^{+0.23}_{-0.23}$ & $34.6^{+5.9}_{-4.9}$ & $0.53^{+0.02}_{-0.02}$ & $59.7^{+4.6}_{-4.8}$ & $0.11^{+0.02}_{-0.02}$ & $1.10$ (2056)\\
\hline
\end{tabular}\\
$^{a}$Column density of the single-column absorption.\\
$^{b}$Column density of the partial-covering absorption.\\
$^{c}$Covering fraction.\\
$^{d}$The highest temperature of the cooling-flow model.\\
$^{e}$Fe abundance.\\
$^{f}$Reduced $\chi^2$ and the number of degree of freedom.\\
\end{minipage}
\end{center}
\end{table*}

\begin{figure*}
\includegraphics[width=0.65\hsize]{figures/broad_band_fit.pdf}
\caption{
Broad-band fit result of Epochs 0-5 of the March 2015 data. Color coding is the same as that of Figure \ref{fig:xis_3rd_spectra_all_intervals}.
The top panel shows the data and the best-fit models with each data set shifted upward by a step of a factor of 2 for better visibility. The middle and the bottom panels show fit residuals of Epochs $0-2$ and Epochs $3-5$, respectively.
}
\label{fig:broad_band_fit}
\end{figure*}

\section{Discussion}\label{sec:discussion}

\subsection{Accretion geometry in quiescent and outburst states}\label{sec:disc_acc_geometry}

The broad-band spectral modeling in outburst showed that the maximum plasma temperature in the post-shock accretion region is almost constant while the hard X-ray flux increased by a factor of $\sim6-7$ from Epoch 0 to Epoch 5 (Tables \ref{tab:xis_nH} and \ref{tab:broad_band_fit}). In the same time period, the absorption column density has increased about three times (Table \ref{tab:broad_band_fit}).
We consider that these increases reflect an enhanced mass accretion rate, and in the present discussion, we adopt the flux enhancement factor as representative of mass transfer rate, because the line-of-sight column density may not have linear dependence on an actual mass accretion rate due to geometrical effect (e.g. \citealt{vrielmannetal2005}). Therefore, it is assumed that about $6-7$ times more mass transfer took place in Epoch 5 than that in Epoch 0.

In magnetic CVs, it is thought that accreting matter is transformed to magnetically-channeled accretion from the accretion disk at a radius where the magnetic pressure balances the gas pressure.
This radius is the so-called magnetospheric radius $R_{\mathrm{A}}$ \citep{warner1995,franketal2002accretionpower}, and scales with the mass accretion rate $\dot{m}$ as $R_{\mathrm{A}}\propto\dot{m}^{-2/7}$.
Since in the current observation period, $\dot{m}$ increased by a factor of $6-7$, and using this scaling, the magnetospheric radius in quiescence ($R_{\mathrm{A,qui}}$) and outburst ($R_{\mathrm{A,out}}$) can be related as $R_{\mathrm{A,out}}\sim0.6~R_{\mathrm{A,qui}}$.

If we assume a WD mass of $M_\mathrm{WD}=0.87~M_\odot$ \citep{morales-ruedaetal2002gkper} and that the quiescent magnetospheric radius should be matched to the co-rotation radius $R_\mathrm{A,qui}=R_\Omega=11.1~R_\mathrm{WD}$ (for $P_{\mathrm{WD}}=351.4$~s), we obtain $R_{\mathrm{A,out}}\sim0.6~R_{\mathrm{A,qui}}=6.6~R_\mathrm{WD}$ when disk-magnetosphere interaction is in equilibrium at an accretion rate of the outburst.
Following the calculations in \citet{suleimanovetal2005}, the maximum plasma temperature in the post-shock accretion region should decrease as
\begin{eqnarray*}
T^{\mathrm{out}}_\mathrm{max} \simeq \left( \frac{1-R_\mathrm{WD}/R_\mathrm{A,out}}{1-R_\mathrm{WD}/R_\mathrm{A,qui}} \right) T^{\mathrm{qui}}_\mathrm{max}
\sim0.93~T^{\mathrm{qui}}_\mathrm{max}.
\end{eqnarray*}
This difference is relatively small compared to the $\sim10\%$ statistical fitting errors associated to the maximum temperature obtained from the broad-band fitting (Table \ref{tab:broad_band_fit}), and therefore it is difficult to definitively discuss the possibility of the temporal variation of the maximum temperature due to the shrunk inner disk radius. However, based on the result that the face values of the maximum temperature do not monotonically decrease throughout the epochs, we may consider that, during the {\it Suzaku}  observation covering about three days after the onset of the outburst, the inner disk radius was rather stable at the similar radius as $R_{\mathrm{A,qui}}$, and probably slowly transiting toward a new equilibrium radius at the increased mass accretion rate. In the previous outburst observation of XY Ari, \citet{hellieretal1997} argued that the disk inner edge transition from $\sim9~R_\mathrm{WD}$ to $\sim4~R_\mathrm{WD}$ took place in about 1~day based on consideration that the inner disk that approached to the WD blocked the view of the lower accretion pole resulting large ($\sim\times10$) pulse amplitude in outburst. \citet{hellieretal2004gkper} mentioned a possibility of similar ``blocking of accretion pole by disk'' for the GK Per outburst. However, at least during the early outburst ($1-3$ days from the onset), our result suggests that the radius of the inner edge of the disk is rather constant and the sinusoidal modulation seen in outburst is solely caused by obscuration by accreting pre-shock gas.

\subsection{Fe fluorescence line shape}
A red-ward wing-like structure that has been moderately detected and reported by \citet{hellierandmukai2004fek} using {\it Chandra}  HETG data in outburst is an interesting topic regarding the Fe fluorescence line.
The authors considered that the red-ward wing could be a Doppler shifted Fe fluorescence line emitted from the pre-shock absorbing gas moves at velocities close to the free-fall velocity (a few times $1000~\mathrm{km}~\mathrm{s}^{-1}$; see also \citealt{hayashietal2011} for a similar study in V1223 Sgr).

In the present study, however, the Fe fluorescence line measured by the XIS can be explained by a single gaussian without any broadening or centroid shift (\S\ref{sec:xis_fe_line}). It will be difficult to detect the reported structure, if exists, due to a degraded CCD energy resolution (FWHM$\sim$200~eV in 2015). In the near future, the {\it ASTRO-H} X-ray microcalorimeter will allow us to measure the line shape in great detail with an energy resolution of FHWM$\sim5$~eV, and to separate the Doppler shifted component from the fluorescence line from the WD surface. If the interpretation of reflection off the pre-shock gas is validated, the Doppler measurement will provide another strong constraint on the accretion geometry in outburst.

\section{Conclusions}
\begin{itemize}
\item {\it Suzaku}  serendipitously observed an onset of the dwarf nova outburst of the intermediate polar GK Per in March 2015. In the latter part of the observation, the flux reached at about 50\% of the maximum of the present outburst (\S\ref{sec:obs}).
\item The WD spin period of $351.4\pm0.5$~s was clearly detected in outburst, and the hard X-ray signals above 16~keV were also modulated at the same period (\S\ref{sec:lc}) suggesting absorption by a dense obscurer with $N_{\mathrm{H}}>10^{23}~\mathrm{cm}^{-2}$.
\item The Fe K fluorescent line intensity increased from $\sim80$~eV to $\sim140$~eV following the onset of the outburst (\S\ref{sec:xis_fe_line}).
\item Multi-temperature spectral model fit to the time-sliced broad-band spectra in $2-60$~keV revealed no significant change of the maximum temperature of the post-shock accreting flow in outburst (\S\ref{sec:spectral_broad}). This may suggest that, although the mass accretion rate increased by $\sim6-7$ times compared to quiescence, the inner accretion disk radius changed little during the observation until $\sim3$~days after the outburst onset.
\end{itemize}

\section*{Acknowledgments}
The authors appreciate the {\it Suzaku}  operation team that made this research possible.
We also acknowledge the use of public data from the Swift data archive.
T.Y. is supported by the Special Postdoctoral Researchers Program in RIKEN.
This work was partly supported by JSPS KAKENHI Grant Number 15K17668.

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