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- matrix polynomial (6)
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- palindromic matrix polynomial (3)
- rank one perturbation (3)
- alternating matrix polynomial (2)
- canonical form (2)
- dissipative Hamiltonian system (2)
- generic perturbation (2)
- matrix pencil (2)
- port-Hamiltonian system (2)
- structured linearization (2)
- symplectic matrix (2)
- Brunovsky form (1)
- H-selfadjoint matrices (1)
- H-symmetric matrices (1)
- Hermitian pencils (1)
- Jacobi algorithm (1)
- Jordan form (1)
- Jordan structure (1)
- Lagrangian invariant subspace (1)
- Matrix triples (1)
- Möbius transformation (1)
- Schur form (1)
- Takagi factorization (1)
- Weyl function (1)
- anti-triangular form (1)
- complex Hamiltonian Jordan form (1)
- complex bilinear forms (1)
- complex skew-symmetric matrix (1)
- complex symmetric matrix (1)
- compound matrix (1)
- conditional stability (1)
- distance to instability (1)
- distance to singularity (1)
- eigenvalue backward error (1)
- elementary divisor (1)
- elementary divisors (1)
- even/odd matrix polynomial (1)
- generic low-rank perturbations (1)
- ingular value decomposition (1)
- invariant factor (1)
- invariant polynomial (1)
- invariant polynomials (1)
- low rank perturbation (1)
- nonlinear eigenvalue problem (1)
- orthogonal matrix (1)
- palindromic (1)
- palindromic QR-algorithm (1)
- palindromic matrix pencil (1)
- perturbation theory (1)
- real distance to instability (1)
- real structured distance to instability (1)
- restricted distance to instability (1)
- restricted real distance to instability (1)
- selfadjoint matrices (1)
- sign characteristic (1)
- singular pencil (1)
- singular pencils (1)
- skew-Hamiltonian matrix (1)
- skew-symmetric (1)
- stability (1)
- structrured distance to instability (1)
- structured Kronecker canonical form (1)
- structured SVD (1)
- structured deflation method (1)
- structured eigenvalue backward error (1)
- structured matrices (1)
- symmetric pencils (1)
- triangularization (1)
Lagrangian invariant subspaces for symplectic matrices play an important role in the numerical solution of discrete time, robust and optimal control problems. The sensitivity (perturbation) analysis of these subspaces, however, is a difficult problem, in particular, when the eigenvalues are on or close to some critical regions in the complex plane, such as the unit circle.
We present a detailed perturbation analysis for several different cases of real and complex symplectic matrices. We analyze stability and conditional stability
as well as the index of stability for these subspaces.
Canonical forms for matrix triples $(A,G,\hat G)$, where
$A$ is arbitrary rectangular and $G$, $\hat G$ are either real symmetric
or skew symmetric, or complex Hermitian or skew Hermitian, are derived.
These forms generalize classical product Schur forms as well as
singular value decompositions.
An new proof for the complex case is given, where there is no need to
distinguish whether $G$ and $\hat G$ are Hermitian or skew Hermitian.
This proof is independent from the results in Bolschakov/Reichstein 1995, where
a similar canonical form has been obtained for the complex case,
and it allows generalization to the real case. Here,
the three cases, i.e., that
$G$ and $\hat G$ are both symmetric, both skew symmetric or one each,
are treated separately.
For selfadjoint matrices in an indefinite inner product, possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation. Genericity is understood in
the sense of algebraic geometry. Special attention is paid to the perturbation
behavior of the sign characteristic. Typically, under such a perturbation,
for every given eigenvalue, the largest Jordan block of the eigenvalue is
destroyed and (in case the eigenvalue is real) all other Jordan blocks
keep their sign characteristic. The new eigenvalues, i.e., those eigenvalues of
the perturbed matrix that are not eigenvalues of the original matrix,
are typically simple, and in some cases information is provided about their sign
characteristic (if the new eigenvalue is real). The main results are proved by using
the well known canonical forms of selfadjoint matrices in an indefinite inner product, a version of the Brunovsky
canonical form and on general results concerning rank one perturbations.
Many applications give rise to matrix polynomials whose coefficients have
a kind of reversal symmetry, a structure we call palindromic.
Several properties of scalar palindromic polynomials are derived,
and together with properties of compound matrices, used to
establish the Smith form of regular and singular T-palindromic matrix polynomials,
over arbitrary fields.
The invariant polynomials are shown to
inherit palindromicity,
and their structure is described in detail.
Jordan structures of palindromic matrix polynomials are characterized,
and necessary conditions for the
existence of structured linearizations established.
In the odd degree case, a constructive procedure for building
palindromic linearizations shows that the necessary conditions are sufficient as well.
The Smith form for *-palindromic polynomials is also analyzed. Finally, results for palindromic matrix polynomials over fields of
characteristic two are presented.
Alternating matrix polynomials, that is, polynomials whose coefficients
alternate between symmetric and skew-symmetric matrices,
generalize the notions of even and odd scalar polynomials.
We investigate the Smith forms of alternating matrix polynomials,
showing that each invariant factor is an even or odd scalar polynomial.
Necessary and sufficient conditions
are derived for a given Smith form to be that of an alternating matrix polynomial.
These conditions allow a characterization of the possible Jordan structures
of alternating matrix polynomials,
and also lead to necessary and sufficient conditions
for the existence of structure-preserving strong linearizations.
Most of the results are applicable to singular as well
as regular matrix polynomials.
We study the perturbation theory of structured matrices under
structured rank one perturbations, and then focus on several classes of complex
matrices. Generic Jordan structures of perturbed matrices are identified.
It is shown that the perturbation
behavior of the Jordan structures
is substantially different from the corresponding theory for unstructured generic
rank one perturbations.
New perturbation results for the behavior of eigenvalues and Jordan forms of real and complex matrices
under generic rank one perturbations are discussed. Several results that are available in the complex
case are proved as well for the real case and the assumptions on the genericity are weakened.
Rank one perturbations that lead to maximal algebraic multiplicities of the ``new" eigenvalues are also
discussed.
We characterize the Smith form of skew-symmetric matrix polynomials
over an arbitrary field $\F$,
showing that all elementary divisors occur with even multiplicity.
Restricting the class of equivalence transformations to unimodular congruences,
a Smith-like skew-symmetric canonical form
for skew-symmetric matrix polynomials is also obtained.
These results are used to analyze the eigenvalue and elementary divisor structure
of matrices expressible as products of two skew-symmetric matrices,
as well as the existence of structured linearizations
for skew-symmetric matrix polynomials.
By contrast with other classes of structured matrix polynomials
(e.g., alternating or palindromic polynomials),
every regular skew-symmetric matrix polynomial
is shown to have a structured strong linearization.
While there are singular skew-symmetric polynomials of even degree
for which a structured linearization is impossible,
for each odd degree we develop a skew-symmetric companion form
that uniformly provides a structured linearization
for every regular and singular skew-symmetric polynomial
of that degree.
Finally, the results are applied to the construction of minimal
symmetric factorizations of skew-symmetric rational matrices.
Canonical forms are developed for several sets of complex matrices that are normal
with respect to an indefinite inner product induced by a nonsingular Hermitian,
symmetric, or skew-symmetric matrix. The most general result covers the case of
polynomially normal matrices, i.e., matrices whose adjoint with respect to the indefinite
inner product is a polynomial of the original matrix. From this result, canonical
forms for matrices that are selfadjoint, skewadjoint, or unitary with respect to the
given indefinite inner product are derived.
Real polynomially normal matrices are studied, i.e., matrices whose adjoint with
respect to the indefinite inner product is a polynomial in the matrix. The set of these matrices is
a subset of indefinite inner product normal matrices that contains all selfadjoint, skew-adjoint, and
unitary matrices, but that is small enough such that all elements can be completely classified. The
essential decomposition of a real polynomially normal matrix is introduced. This is a decomposition
into three parts, one part having real spectrum only and two parts that can be described by two
complex matrices that are polynomially normal with respect to a sesquilinear and bilinear form,
respectively. In the paper, the essential decomposition is used as a tool in order to derive a sufficient
condition for existence of invariant semidefinite subspaces and to obtain canonical forms for real
polynomially normal matrices. In particular, canonical forms for real matrices that are selfadjoint,
skewadjoint, or unitary with respect to an indefinite inner product are recovered.