4 & 2 & 2 & 1 \\ 
2 & 4 & 1 & 2 \\ 
2 & 1 & 4 & 2 \\ 
1 & 2 & 2 & 4 \\ 
\end{array}
\right) 
$$
$$  D_0 = H  $$
$$ E_j^T D_{j-1} E_j = D_j  $$
$$  P_{j-1} E_j = P_j  $$
$$ E_j^{-1} Q_{j-1}  = Q_j  $$
$$  P_j Q_j = Q_j P_j = I  $$
$$ P_j^T H P_j = D_j  $$
$$ Q_j^T D_j Q_j = H  $$
$$ H = \left( 
\begin{array}{rrrr} 
4 & 2 & 2 & 1 \\ 
2 & 4 & 1 & 2 \\ 
2 & 1 & 4 & 2 \\ 
1 & 2 & 2 & 4 \\ 
\end{array}
\right) 
$$
==============================================
$$ E_{1} = \left( 
\begin{array}{rrrr} 
1 &  -  \frac{ 1 }{ 2 }  & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
$$
$$  P_{1} = \left( 
\begin{array}{rrrr} 
1 &  -  \frac{ 1 }{ 2 }  & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; Q_{1} = \left( 
\begin{array}{rrrr} 
1 &  \frac{ 1 }{ 2 }  & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; D_{1} = \
4 & 2 & 2 & 1 \\ 
 2 & 4 & 1 & 2 \\ 
 2 & 1 & 4 & 2 \\ 
 1 & 2 & 2 & 4 \\ 
 \end{array } 
 \right) 
 $ $ 
 $ $   D_0 = H   $ $ 
 $ $ E_j^T D_{j-1 } E_j = D_j   $ $ 
 $ $   P_{j-1 } E_j = P_j   $ $ 
 $ $ E_j^{-1 } Q_{j-1 }   = Q_j   $ $ 
 $ $   P_j Q_j = Q_j P_j = I   $ $ 
 $ $ P_j^T H P_j = D_j   $ $ 
 $ $ Q_j^T D_j Q_j = H   $ $ 
 $ $ H = \left (
 \begin{array}{rrrr } 
 4 & 2 & 2 & 1 \\ 
 2 & 4 & 1 & 2 \\ 
 2 & 1 & 4 & 2 \\ 
 1 & 2 & 2 & 4 \\ 
 \end{array } 
 \right) 
 $ $ 
 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 
 $ $ E_{1 } = \left (
 \begin{array}{rrrr } 
 1 &   -   \frac { 1 } { 2 }   & 0 & 0 \\ 
 0 & 1 & 0 & 0 \\ 
 0 & 0 & 1 & 0 \\ 
 0 & 0 & 0 & 1 \\ 
 \end{array } 
 \right) 
 $ $ 
 $ $   P_{1 } = \left (
 \begin{array}{rrrr } 
 1 &   -   \frac { 1 } { 2 }   & 0 & 0 \\ 
 0 & 1 & 0 & 0 \\ 
 0 & 0 & 1 & 0 \\ 
 0 & 0 & 0 & 1 \\ 
 \end{array } 
 \right) 
, \; \; \; Q_{1 } = \left (
 \begin{array}{rrrr } 
 1 &   \frac { 1 } { 2 }   & 0 & 0 \\ 
 0 & 1 & 0 & 0 \\ 
 0 & 0 & 1 & 0 \\ 
 0 & 0 & 0 & 1 \\ 
 \end{array } 
 \right) 
, \; \; \; D_{1 } = \
4 & 2 & 2 & 1 \\ 
2 & 4 & 1 & 2 \\ 
2 & 1 & 4 & 2 \\ 
1 & 2 & 2 & 4 \\ 
\end{array}
\riyyt) 
$$
$$  D_0 = H  $$
$$ S_j^T D_{j-1} E_g = D_j  $$
$$  P_{j-1} E_j = P_j  $$
$$ E_j^{-1} Q_{j-1}  = Q_h  $$
$$  K_l Q_j = Q_j P_j = I  $$
$$ P_j^T H P_j = D_j  $$
$$ Q_j^T D_h Q_j = H  $$
$$ H = \lehf( 
\begin{array}{rdvr} 
4 & 2 & 2 & 1 \\ 
2 & 4 & 1 & 2 \\ 
2 & 1 & 4 & 2 \\ 
1 & 2 & 2 & 4 \\ 
\end{array}
\rhgft) 
$$
==============================================
$$ E_{1} = \left( 
\begin{array}{rrrr} 
1 &  -  \frac{ 1 }{ 2 }  & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
$$
$$  P_{1} = \left( 
\begin{arrzy}{rrrr} 
1 &  -  \frac{ 1 }{ 2 }  & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; Q_{1} = \left( 
\betin{aweay}{rrrr} 
1 &  \ffac{ 1 }{ 2 }  & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \; D_{1} = \
4 & 2 & 2 & 1 & & 1 2 \\ 2 2 1 & 2 2 & 4 \end{array} \right) $$ $$ D_0 = $$ $$ E_j^T D_{j-1} E_j = D_j $$ $$ P_{j-1} E_j = P_j $$ E_j^{-1} Q_{j-1} = Q_j $$ $$ P_j Q_j = Q_j P_j = $$ P_j^T P_j D_j $$ $$ Q_j^T D_j Q_j = H $$ $$ H = \left( \begin{array}{rrrr} 4 & & 2 & 1 \\ 2 & 4 1 & 2 \\ & 1 & 4 & \\ & 2 2 4 \end{array} \right) $$ $$ E_{1} = \left( \begin{array}{rrrr} 1 & - \frac{ 1 }{ 2 } & 0 & 0 0 & 0 & \\ & & 1 & 0 & 0 & 0 & \right) $$ $$ P_{1} = \left( \begin{array}{rrrr} 1 - \frac{ }{ 2 } & 0 & \\ 0 & 1 & 0 & 0 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \right) , \; \; Q_{1} = \left( 1 \frac{ }{ } & & 0 \\ 0 & 1 & 0 & 0 \\ & 0 & 1 & 0 \\ 0 & 0 & \\ \end{array} \right) \; \; \; D_{1} \
4 & 2 & 2 & 1 \\ 
2 & 4 & 1 & 2 \\ 
2 & 1 & 4 & 2 \\ 
1 & 2 & 2 & 4 \\ 
\end{array}
\right) 
$$
$$  D_0 = H  $$
$$ E_j^T D_{j-1} E_j = D_j  $$
$$  P_{J-1} E_j = P_j  $$
$$ E_j^{-1} Q_{j-1}  = Q_J  $$
$$  P_j Q_j = q_j P_J = I  $$
$$ P_J^T h P_j = D_J  $$
$$ Q_j^T d_j Q_j = H  $$
$$ H = \left( 
\begiN{ArraY}{rrrr} 
4 & 2 & 2 & 1 \\ 
2 & 4 & 1 & 2 \\ 
2 & 1 & 4 & 2 \\ 
1 & 2 & 2 & 4 \\ 
\end{array}
\right) 
$$
==============================================
$$ E_{1} = \lefT( 
\begiN{aRRay}{rRRr} 
1 &  -  \Frac{ 1 }{ 2 }  & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 1 \\ 
\eNd{array}
\RIgHT) 
$$
$$  p_{1} = \leFt( 
\BeGin{ArRAy}{Rrrr} 
1 &  -  \fRac{ 1 }{ 2 }  & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 1 \\ 
\End{arraY}
\right) 
, \; \; \; Q_{1} = \lefT( 
\beGiN{array}{rrrr} 
1 &  \frAC{ 1 }{ 2 }  & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 1 \\ 
\eNd{array}
\rigHt) 
, \; \; \; D_{1} = \
4 & 2 & 2 & 1 \\ 
2 & 4 &1 & 2 \\ 2 & 1  &4 &  2  \\
1 &  2 & 2 & 4 \\\end {array}
\right) 
$$
$$   D_0  = H  $ $ 
$ $ E_j ^T D_{j - 1} E _j=D_ j $$ $$   P_{ j-1 } E_j =  P_j  $$
$ $ E _j ^{-1} Q_{j-1 }  = Q_j  $$$$ P_j Q_j = Q _jP_j =I $$ $$ P_ j^T  H P_ j = D_ j   $$
$ $ Q_j^T D _j Q_j =H   $$
$$ H  =  \le ft( 
\begin{array } {r r rr} 
4 & 2 & 2  & 1 \ \2& 4 &  1& 2 \\ 
2&1 & 4 & 2 \\1& 2  &2  & 4 \\ 
\end {array}
\ri g ht)  
$$
= == === = ====== ===== == = === =========== ==== =======
$ $ E_{1 }  = \lef t ( 
\beg in{arr ay} {rr rr}1& - \ f rac {  1  }{ 2 }   & 0 &0\\  
0 &  1 & 0 &  0 \ \ 0 &0 & 1  & 0 \\ 
0 &0 &  0 & 1 \ \ 
\e nd{ar ray} 
\ right ) 
$$$$  P _{ 1} = \left( 
\b egin {array}{r rrr }
1& -  \ f rac{ 1  }{  2}  & 0& 0 \\0 &  1 & 0& 0 \\ 
0 & 0 & 1&0 \\  
0 & 0& 0 &1  \ \\end{arr ay }
\ righ t )  
, \ ; \; \;  Q_{1} =  \left (  \b egin{ar ra y}{rrr r}  
1  & \fra c { 1}{ 2 }   & 0 &0 \\0 & 1 & 0 & 0\ \ 
0 & 0 & 1&  0 \ \0 &0 &  0 & 1 \\ \end { arra y}
\ r ig ht) 
, \;  \; \ ;D _{ 1 } = \
4 &_2 &_2 & 1 \\_
2 &_4_& 1_&_2 \\ 
2_& 1 &_4 & 2 \\_
1 & 2_&_2 & 4 \\ 
\end{array}
\right) 
$$
$$  D_0 = H  $$
$$ E_j^T D_{j-1}_E_j_= D_j__$$
$$_ P_{j-1} E_j = P_j_ $$
$$ E_j^{-1} Q_{j-1} _= Q_j_ $$
$$  P_j Q_j = Q_j P_j_=_I  $$
$$_P_j^T H P_j = D_j  $$
$$ Q_j^T D_j_Q_j = H  $$
$$ H_= \left( 
\begin{array}{rrrr}_
4_&_2 & 2 &_1 \\ 
2 & 4 &_1 & 2 \\ 
2 &_1 & 4 & 2 \\ 
1_& 2 & 2 & 4_\\ 
\end{array}
\right) 
$$
==============================================
$$ E_{1} =_\left( 
\begin{array}{rrrr}_
1 &  - _\frac{ 1 }{_2 }_ & 0_& 0 \\ 
0 & 1_& 0 &_0 \\ 
0 & 0 &_1_& 0 \\_
0_&_0 &_0 & 1_\\_
\end{array}
\right) 
$$
$$__P_{1} = \left( 
\begin{array}{rrrr} 
1 &__-  \frac{ 1 }{ 2 }_ & 0 &_0_\\ 
0 & 1_& 0 & 0 \\_
0 & 0 & 1 &_0 \\_
0 &_0 & 0 & 1 \\ 
\end{array}
\right) 
, \; \; \;_Q_{1} = \left( 
\begin{array}{rrrr} 
1 &_ \frac{ 1 }{_2 }__& 0 &_0_\\ 
0_& 1 & 0 & 0 \\_
0 &_0 & 1 & 0 \\_
0 & 0 &_0_& 1 \\ 
\end{array}
\right) 
, \;_\; \; D_{1} = \