On the Perturbation Theory for Unitary Eigenvalue Problems.
Some aspects of the perturbation theory for eigenvalues of unitary matrices are considered. Making use of the close relation between unitary and Hermitian eigenvalue problems a Courant--Fischer-type theorem for unitary  matrices is derived and an inclusion theorem analogous to the  Kahan theorem for  Hermitian matrices is presented. Implications for the special case of unitary Hessenberg matrices are discussed.
Introduction
. New numerical methods to compute eigenvalues of unitary
matrices have been developed during the last ten years. Unitary QR-type methods
[19, 9], a divide-and-conquer method [20, 21], a bisection method [10], and some special
methods for the real orthogonal eigenvalue problem [1, 2] have been presented.
Interest in this task arose from problems in signal processing [11, 29, 33], in Gaussian
quadrature on the unit circle [18], and in trigonometric approximations [31, 16] which
can be stated as eigenvalue problems for unitary matrices, often in Hessenberg form.
As those numerical methods exploit the rich mathematical structure of unitary ma-
trices, which is closely analogous to the structure of Hermitian matrices, the methods
are efficient and deliver very good approximations to the desired eigenvalues.
There exist, however, only a few perturbation results for the unitary eigenvalue
problem, which can be used to derive error bounds for the computed eigenvalue ap-
proximations. A thorough and complete treatment of the perturbation aspects associated
with the numerical methods for unitary eigenvalue problems is still missing.
The following perturbation results have been obtained so far. If U and e
U are
unitary matrices with spectra
respectively, we can
arrange the eigenvalues in diagonal matrices   and e
, respectively, and consider as a
measure for the distance of the spectra
d  (oe(U); oe( e
U
where the minimum is taken over all permutation matrices P and the norm is either
the spectral or the Frobenius norm. By the Hoffman-Wielandt theorem (see, e.g.,[34])
we get
dF (oe(U); oe( e
U
U
Bhatia and Davis [5] proved the corresponding result for the spectral norm
U
Schmidt, Vogel & Partner Consult, Gesellschaft fur Organisation und Managementberatung
mbH, Gadderbaumerstr. 19, 33602 Bielefeld
y Universitat Bremen, Fachbereich 3 - Mathematik und Informatik, 28334 Bremen, Germany,
email: angelika@math.uni-bremen.de
z Universitat Bremen, Fachbereich 3 - Mathematik und Informatik, 28334 Bremen, Germany,
email: heike@math.uni-bremen.de
Elsner and He consider a relative error in [15]. They use the measure
e
d (oe(U); oe( e
U
where again They prove that
e
d (oe(U); oe( e
U
U)jj
is the Cayley transformation of U (assuming here
that
To each eigenvalue  of U , where \Gamma1 62 oe(U ), we can associate an angle '  by
defining
with \Gamma=2  '  ! =2. It is the angle formed by the line from -1 through  and the
real axis (see also Section 2). With respect to their angles the eigenvalues of U and
e
U have a natural ordering on the unit circle. Elsner and He give sine- and tangent-
interpretations of the above inequality in terms of these angles. Furthermore they
show that with respect to a certain cutting point i on the unit circle the eigenvalues
of U and e
U have a natural ordering f j (i)g and f e  j (i)g on the unit circle such that
An interlacing theorem for unitary matrices is also presented in [15], showing that the
eigenvalues of suitably modified principal submatrices of a unitary matrix interlace
those of the complete matrix on the unit circle (see Section 2).
In this paper we consider further aspects of the perturbation problem for the
eigenvalues of a unitary matrix U . In Section 2 we show how the angles i are related
to the eigenvalues of the Cayley transform of U . With the aid of this relation we
can give a min-max-characterization for the angles of U 's eigenvalues in analogy to
the Courant-Fischer theorem for Hermitian matrices. We also show that tangents of
these angles can be characterized by usual Rayleigh quotients corresponding to the
generalized eigenvalue problem
Furthermore we prove a Kahan-like inclusion theorem showing that the eigenvalues of
a certain modified leading principal submatrix of U determine arcs on the unit circle
such that each arc contains an eigenvalue of U . In applications unitary matrices are
often of Hessenberg form. In Section 3 we recall that a unitary unreduced Hessenberg
matrix H has a unique parameterization reflection
parameters
H completely. We show the implications of the results in Section 2 for the special
case of unitary Hessenberg matrices. In particular it will be seen that the modified
kth leading principal submatrix in this special case is just H(fl
We discuss the dependence of the eigenvalues on this last reflection parameter
i. Finally Section 4 will give numerical examples which elucidate the statements
proved in Section 3.
2. Perturbation Results for unitary Matrices. Unitary matrices have a rich
mathematical structure that is closely analogous to that of Hermitian matrices. In
this section we first discuss the intimate relationship between unitary and Hermitian
matrices which indicates that one can hope to find unitary analogues for the good numerical
methods and for the theoretical results that exist for the symmetric/Hermitian
eigenvalue problem. We will adapt some eigenvalue bounds for Hermitian matrices to
the unitary case.
Let ae be a complex unimodular number. The Cayley transformation with respect
to ae maps the unitary matrices whose spectrum does not include ae, onto the Hermitian
matrices. The Cayley transformation with respect to ae for a unitary matrix U 2 C n\Thetan
is defined as
where ae is not an eigenvalue of U and
\Gamma1. I n denotes the n \Theta n identity matrix.
A simple calculation shows that C(U) is Hermitian. The mapping is one-to-one and
the inverse Cayley transformation with respect to ae for a Hermitian matrix X is given
by
The symmetric/Hermitian eigenproblem has been extensively studied, see, e.g. [28,
17, 24, 26]. Due to this relation between Hermitian and unitary matrices, one can
hope to get similar results for unitary matrices.
With the aid of the Cayley transformation we can order the eigenvalues con-
veniently. Let  be the eigenvalues of U numbered starting at ae moving
counterclockwise along the unit circle. Let  1   be the eigenvalues of
For simplicity assume that ae = \Gamma1. Then
Cnf0g the argument of z, arg(z) 2 (\Gamma; ] is defined
by
arctan( Im(z)
arctan( Im(z)
The Cayley transformation of  k is the tangent of the angle '
is formed by the real axis and the straight line through  k and \Gamma1:
l k
Hence it is reasonable to define
This also gives a complete ordering of the points on the unit circle with respect to
the cutting point \Gamma1. Note that the complete ordering excludes the cutting point \Gamma1.
For a different cutting point the orders of the eigenvalues are only changed cyclically.
are complex unimodular numbers such that
denote the open arc from the point i 1 to the point i 2 on the unit circle (moving
counterclockwise).
The Courant-Fischer theorem (see, e.g., [17, Theorem 8.1.2]) characterizes the
eigenvalues of Hermitian matrices by Rayleigh quotients. A similar characterization
can be given for the eigenvalues of unitary matrices. Let U 2 C n\Thetan be a unitary
matrix with eigenvalues  . Assume that \Gamma1 is not an eigenvalue of U and
number the eigenvalues starting at \Gamma1 moving counterclockwise along the unit circle.
vn be an orthonormal basis in C n\Thetan of eigenvectors of U . Let z 2 C n with
1. Then we can expand z as
From
Because of
we see that the Rayleigh quotient z H Uz lies in the convex polygon which is spanned
by the eigenvalues of U .
l 1
l 2
l 3
l 4
l 5
z H Uz
Theorem 2.1. With the notation given above we obtain for
min
denotes the set of all k-dimensional subspaces of C n .
In particular,
Proof. Let be an orthonormal basis of eigenvectors of U . Let V 2
Vn\Gammak+1 . Then z be a vector in this intersection, jjzjj
Then
Hence, the Rayleigh quotient z H Uz lies in the convex polygon spanned by the eigen-values
Therefore
and
min
Now consider the subspace of dimension which is spanned by v
vector z; jjzjj this subspace can be written as
Hence, the Rayleigh quotient z H Uz lies in the convex polygon spanned by the eigen-values
Therefore
and
min
The second equation can be shown analogously.
Corollary 2.2. With the notation given above we define for z 2 C n with
Then
and for
min
Proof. A simple calculation yields (2.2). The rest of the corollary follows from
Theorem 2.1 and the monotonicity of the function tan in (\Gamma
The corollary shows that the angles ' k can be characterized by usual Rayleigh
quotients. R(z) can be interpreted as the Rayleigh quotient corresponding to the
generalized eigenvalue problem
Since U is unitary, -(I n +U H )(I n \Gamma U) is Hermitian and (I n +U H )(I n +U) is Hermitian
and positive definite. (2.3) is equivalent to the eigenvalue problem
for the Cayley transformation of U .
Remark 2.3. For ease of notation the above theorem and corollary are formulated
for the case that ae = \Gamma1 is not an eigenvalue of U . This restriction is not necessary,
one can proof the corresponding statements for any cutting point ae 2 C; not
an eigenvalue of U .
In [15], the Cauchy interlacing theorem for Hermitian matrices is generalized to
the unitary case. The Cauchy interlace theorem shows that the eigenvalues of the k \Theta k
leading principal submatrix of a Hermitian matrix X interlace the eigenvalues of X .
Adapting this theorem to the unitary case, one has to deal with the problem that
leading principal submatrices of unitary matrices are in general not unitary and that
their eigenvalues lie inside the unit circle. In [15] it is shown that certain modified
leading principal submatrices of a unitary matrix U have the property that their
eigenvalues interlace with those of U .
Theorem 2.4. [15, Theorem 5.2 and 5.3] Let
U 11 U 12
U
be an n \Theta n unitary matrix, U 11 the k \Theta k leading principal submatrix of U , and
ae is not an eigenvalue of U . Then U k is unitary. Let
be the eigenvalues of U and
be those of U k ordered with respect to ae. Then
U k is called the modified kth leading principal submatrix of U . Furthermore,
analogues of the Hoffman-Wielandt theorem and a Weyl-type theorem are derived in
[15].
With the help of Theorem 2.4 one can specify for each eigenvalue of a modified
leading principal submatrix U k an arc on the unit circle which contains that eigenvalue.
These bounds are fairly rough, especially if k is much smaller than n. This result is of
theoretical nature, because in practice we are more interested in the question, whether
the arc contains an eigenvalue of U or not.
The same problem arises in the Hermitian case. Lehmann and Kahan derived
inclusion theorems which consider this problem (see, e.g., [28] and the references
therein). A special case of their results is
Theorem 2.5. [28, Theorem on page 196] Let X 2 C n\Thetan be a Hermitian matrix.
Partition X as
Let  i be the eigenvalues of X
then each interval [ contains an eigenvalue of X, where
The following theorem states the analogous result for unitary matrices.
Theorem 2.6. Let U 2 C n\Thetan be a unitary matrix and let U be partitioned as
in (2.4). Let ae 2 C; be not an eigenvalue of U . Define a unitary matrix
U k as in (2.5) with eigenvalues  . The eigenvalues are numbered starting
at ae moving counterclockwise along the unit circle. If rank(U 21 then each arc
on the unit circle, contains at least one eigenvalue of U , where
Proof. Since ae is not an eigenvalue of U , it is not an eigenvalue of U 22 : Assume
ae is an eigenvalue of U 22 . Then there is a normalized eigenvector x 2 C n\Gammak such that
U 22
U
x
U 22 x
aex
As and U is unitary, U 12 x has to be zero. But this would imply that
ae is an eigenvalue of U in contradiction to our general assumption. Hence ae is not an
eigenvalue of U 22 .
U k is defined as
can be interpreted as the Schur complement of U 22 \Gamma aeI n\Gammak in U \Gamma aeI n .
We can make use of this fact to construct (U \Gamma aeI using the following result of
Duncan [13] (see Corollary 2.4 in [27])
Let A 2 C n\Thetan be partioned as
Let A and H be nonsingular, then is the
Schur complement of H in A, T is nonsingular and
\GammaH
We obtain
In particular, (U is the k \Theta k leading principal submatrix of (U \Gamma aeI
Now we consider the Cayley transformation with respect to ae of U
This yields
We partition X as we did U :
is the k \Theta k leading principal submatrix of X . From (2.6) it follows that
Therefore, X k is the Cayley transformation of U k . Further we obtain from (2.6):
If rank(U 21 as the other two matrices in the product have
full rank.
Now we can use Theorem 2.5 to obtain that each interval formed by two eigen-values
of X k contains at least one eigenvalue of X . We have seen in (2.1) that the
eigenvalues of X and X k can be obtained from those of U and U k via the Cayley trans-
formation. As the Cayley transformation is monotone, this yields: each arc
on the unit circle, contains at least one eigenvalue of U . For the two
outer arcs (ae;  1 ae) the statement follows directly from Theorem 2.4.
The last result we mention in this section clarifies the question of how the eigen-values
of a unitary matrix change if the matrix is modified by a unitary differing from
I only by rank one. For the Hermitian case, the answer is given, e.g., in [17, chapter
12.5.3]. For the unitary case we obtain
Theorem 2.7. Let U; S 2 C n\Thetan be unitary matrices and S such that
Then the eigenvalues of U and US interlace on the unit circle.
Proof. See [4, section 6].
3. Unitary upper Hessenberg Matrices. It is well known that any (unitary)
n \Theta n matrix can be transformed to an upper Hessenberg matrix H by a unitary
similarity transformation Q. If the first column of Q is fixed and H is an unreduced
upper Hessenberg matrix with positive subdiagonal elements (that is h i+1;i ? 0),
then the transformation is unique. Any n \Theta n unitary upper Hessenberg matrix
with nonnegative subdiagonal elements can be uniquely parameterized by
parameters. This compact form is used in [1, 3, 9, 11, 14, 19, 20, 21, 22, 23, 32] to
develop fast algorithms for solving the unitary eigenvalue problem.
Let
with
e
with
The product
oe
is a unitary upper Hessenberg matrix with positive subdiagonal elements. Conversely,
n\Thetan is a unitary upper Hessenberg matrix with positive subdiagonal elements,
then it follows from elementary numerical linear algebra that one can determine matrices
Gn such that e
I . Since H as a unitary
matrix has a unique inverse, this has to be e
1 . Thus H has a unique
factorization of the form
Gn (fl n
The Schur parameters ffl k g n
k=1 and the complementary Schur parameters foe k g n
can
be computed from the elements of H by a stable O(n 2 ) algorithm [19]. In statistics
the Schur parameters are referred to as partial correlation coefficients and in signal
processing as reflection coefficients [2, 11, 12, 25, 29, 30, 33].
If oe we have the direct sum decomposition
Hence, in general oe 1 oe 2 :::oe assumed if the factorization (3.1) is used to
solve a unitary eigenvalue problem. Such a unitary upper Hessenberg matrix is called
unreduced. If  is an eigenvalue of an unreduced Hessenberg matrix, then its geometric
multiplicity is one [17, Theorem 7.4.4]. Since unitary matrices are diagonalizable, no
eigenvalue of an unreduced unitary upper Hessenberg matrix is defective, that is, the
eigenvalues of an unreduced unitary upper Hessenberg matrix are distinct.
We will adapt the general theorems given in the last section to the more specific
case of unitary upper Hessenberg matrices. Let H 2 C n\Thetan be a unitary upper Hessenberg
matrix with positive subdiagonal elements,
as
where H 11 is the k \Theta k leading principal submatrix. From Theorem 2.4 we obtain that
the modified kth leading principal submatrix of H , H
is unitary (if ae 2 C; not an eigenvalue of H). As H k is a unitary upper
Hessenberg matrix, we can factor H
Taking
a closer look at H k reveals that H k differs from H 11 only in the last column. Hence the
modification of H 11 to H k is equivalent to a modification of the reflection coefficient
The following theorem by Bunse-Gerstner and He characterizes the correct choice of
the parameter
Theorem 3.1. Let n\Thetan be a unitary upper Hessenberg
matrix with positive subdiagonal elements. For
as in (3.2). Let ae 2 C; not be an eigenvalue of H. Define parameters i l (ae);
(3.
Then
be the eigenvalues of H k (fl
respectively, where the eigenvalues are numbered starting at ae moving counterclockwise
along the unit circle. Then for each the eigenvalue  i lies on the arc
Proof. The interlace property follows directly from Theorem 2.4. For the rest of
the proof see [10].
Using Theorem 2.6 and 3.1 we obtain
Theorem 3.2. Let n\Thetan be a unitary upper Hessenberg
matrix with positive subdiagonal elements. Let ae 2 C; not be an eigenvalue
of H. For ng let H is defined as
in (3.3). Let  be the eigenvalues of H k . Then for each arc
on the unit circle contains at least one eigenvalue of H, where
ae.
Moreover,we obtain
Theorem 3.3. Let n\Thetan be a unitary upper Hessenberg
matrix with positive subdiagonal elements. Then for any i 2 C; there exists a
cutting point ae 2 C; such that the eigenvalues of H
H have the interlace properties with respect to ae on the unit circle given by Theorem
3.1 and 3.2.
Proof. We will show that ae 7! i k (ae) is an automorphism on the unit circle, this
proves the theorem. Note that for unitary upper Hessenberg matrices with positive
subdiagonal elements we have jfl
Obviously
ae 7! i
is bijective on the unit circle. The same is true for the mapping
2:
Hence ae 7! i k (ae) is a one-to-one mapping of the unit circle onto itself.
The statement of the above theorem can be summarized as follows: Any leading
principal submatrix of a unitary upper Hessenberg matrix with positive lower
subdiagonal elements can be modified to be unitary by replacing the last reflection
coefficient with a parameter on the unit circle. No matter how this parameter is
chosen, there is always a cutting point ae on the unit circle such that the eigenvalues
of the modified leading principal submatrix and those of the entire matrix satisfy the
interlace properties given by Theorem 3.1 and 3.2.
Disregarding the cutting point ae and the two arcs formed with it, Theorem 3.3 implies
the following corollary.
Corollary 3.4. Let n\Thetan be a unitary upper Hessenberg
matrix with positive subdiagonal elements. For
ng. Then every arc on the unit circle formed
by two eigenvalues of H k contains an eigenvalue of H.
In particular the above theorems show that the eigenvalues of two consecutive
modified leading principal submatrices H k and H k+1 of a unitary upper Hessenberg
matrix with positive subdiagonal elements interlace on the unit circle. More specifi-
cally, consider the modified leading principal submatrices H
and H n. The eigenvalues
of H k and H k+1 interlace with respect to the cutting point ae on the unit circle
where ae is given by
The remaining question is: how strongly do the eigenvalues of H k (fl
depend of the choice of i? We present some results on this dependence on the last
reflection parameter.
Theorem 3.5. Let H a
upper Hessenberg matrices with positive subdiagonal elements, ji a
1. The eigenvalues of H a and H b interlace on the unit circle.
2. (H a where the eigenvalue variation (U; B) is defined by
i2f1;:::;ng
permutation of
the  i 's being the eigenvalues of U and the  i 's those of B.
3. Let  a
n and  b
n be the eigenvalues of H a and H b . Let
be the Schur decomposition of H a , S
i;j=1 . Then for
min
a
ji a
Proof.
1. We have
S is unitary and rank(In \Gamma 1. According to Theorem 2.7, the eigenvalues
of H a and H b interlace on the unit circle.
2. As the matrices H a and H b differ only in the last column we have
Gn (i a
Since H a and H b are unitary, the statement 2: follows from the following
theorem of Bhatia/Davis [5]:
For all constant multiplies of two unitary
matrices Q and V we have
(For a completely different proof and extension of the result to multiples of
unitaries see [6]. When U and B are Hermitian, the above inequality is a
classical result of Weyl).
3. H b is unitary and therefore unitarily diagonalizable. The first inequality
follows directly from the following easy to prove result [8, Satz 1.8.14]:
Let A 2 C n\Thetan be diagonalizable,
and
Then
min
i2f1;:::;ng
Furthermore we obtain
Gn (i a
Hence, eigenvalues of a unitary upper Hessenberg matrix, whose eigenvectors have
a small last component, are not sensitive to changes in the last reflection parameter.
4. Numerical Examples. Numerical experiments are presented to elucidate
the statements of Section 3. The eigenvalues of a unitary upper Hessenberg matrix
H are compared with the eigenvalues of modified kth leading principal submatrices
H k for different dimensions k. The essential statements of Section 3 can be observed
clearly:
ffl Between two eigenvalues of H k on the unit circle there lies an eigenvalue of
(Corollary 3.4).
ffl The eigenvalues of unitary upper Hessenberg matrices, whose corresponding
eigenvectors have a small last component, are not sensitive against changes
of the last reflection coefficient. (Theorem 3.5).
All computations were done using MATLAB 1 on a SUN SparcStation 10.
A unitary upper Hessenberg matrix constructed
from 20 randomly chosen reflection coefficients C. The eigenvalues
j of H lie randomly on the unit circle. The eigenvalue  j of the modified kth
leading principal submatrices H were computed for different
dimensions
For the first example
was chosen. The eigenvalues of H and H k are
plotted for in the following typical figure. The eigenvalues of H
are marked by 'o', the eigenvalues of H k by '*'.
1 MATLAB is a trademark of The MathWorks, Inc.
For the second example a random complex number chosen. The
following figure displays the same information as before.
Corollary 3.4 states that every arc on the unit circle formed by two eigenvalues of
contains an eigenvalue of H . This can be seen in the above figures. Comparing the
results of the two examples presented, one observes that independent of the choice
of i k the same eigenvalues of H are approximated. In Theorem 3.5 it was proven
that if the last component of an eigenvector of a unitary upper Hessenberg matrix is
small, then the corresponding eigenvalue is not sensitive against changes in the last
reflection coefficient. Individual bounds for the minimal distance of each eigenvalue
a
of H to the eigenvalues  b
are given
min
j2f1;:::;ng
a
is the 'th component of the eigenvector for the ith eigenvalue
of H a . This means that if there is an eigenvalue  a
of H a such that the last component
of the corresponding eigenvector is small, then any unitary upper Hessenberg matrix
of the form H b will have an eigenvalue  b
j that is close to  a
The following table reports the minimal distance between each eigenvalue  fl k
i of
and the eigenvalues  rand
(where randomly chosen as above) as well as the error bounds for
10. The absolute difference between i a and i b was ji a
min
j2f1;:::;ng
6 2.4628e-03 7.1265e-02
9 8.4488e-03 6.0709e-01
Comparing the actual minimal distance with the error bound one observes that the
approximations are much better than the error bound predicts.
The same results can be observed for larger unitary upper Hessenberg matrices
H . Moreover, one can observe that the eigenvalues of the modified leading principal
submatrices H 0
k interlace on the unit circle with respect to a cutting point
ae.
5. Concluding Remarks. In this paper, we have proved that the angles ' k
associated with the eigenvalues  j of a unitary matrix U can be characterized by
Rayleigh quotients. An inclusion theorem for the eigenvalues of symmetric matrices
given by Kahan was adapted to the unitary case. We discussed the special case of
unitary Hessenberg matrices, which is important for certain applications. We proved
that every arc on the unit circle formed by two eigenvalues of a modified kth leading
principal submatrix of a unitary upper Hessenberg matrix contains an eigenvalue of
the complete matrix. Results on the dependence of the eigenvalues of unitary upper
Hessenberg matrices on the last reflection coefficient are given.
Parts of this paper (Section 2 and most of Section first appeared in [7]. Bohn-
horst analyses the connection between a unitary matrix U and its Cayley transformation
more closely with the help of structure ranks.

Acknowledgments

. All three authors owe a special thanks to Ludwig Elsner
for stimulating discussions and many helpful suggestions.



--R

On the Eigenproblem for Orthogonal Matrices

An Implementation of a Divide and Conquer Algorithm for the Unitary Eigenproblem
On the spectral decomposition of Hermitian matrices modified by low rank perturbations with applications
A. bound for the

Beitrage zur numerischen Behandlung des unitaren Eigenwertproblems

Schur Parameter Pencils for the Solution of the Unitary Eigenproblem
On a Sturm sequence of polynomials for unitary Hessenberg matrices
Computing Pisarenko Frequency Estimates
Speech modelling and the trigonometric moment problem
Some devices for the solution of large sets of simultaneous linear equations (with an appendix on the reciprocation of partioned matrices)
Global convergence of the QR algorithm for unitary matrices with some results for normal matrices
Perturbation and Interlace Theorems for the Unitary Eigenvalue Prob- lem

Matrix Computation
Positive Definite Toeplitz Matrices

A Divide and Conquer Algorithm for the Unitary Eigenproblem

Convergence of the Shifted QR Algorithm for Unitary Hessenberg Matrices

Matrix Analysis
A Tutorial Review
A Survey of Matrix Theory and Matrix Inequalities
Schur complements and statistics
The Symmetric Eigenvalue Problem
The Retrieval of Harmonics from a Covariance Function
Fast Approximation of Dominant Harmonics by Solving an Orthogonal Eigenvalue Problem
Discrete Least Squares Approximation by Trigonometric Polynomials
Bestimmung der Eigenwerte orthogonaler Matrizen
Duality theory of composite sinusoidal modelling and linear prediction
The algebraic eigenvalue problem
--TR
unitary eigenvalue problem;perturbation theory