15A18 Eigenvalues, singular values, and eigenvectors
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- eigenvalues (4)
- matrix polynomial (3)
- nonlinear eigenvalue problem (3)
- $\mu$-values (2)
- Hamiltonian matrix (2)
- Smith form (2)
- backward error (2)
- condition numbers (2)
- matrix pencil (2)
- palindromic eigenvalue problem (2)
In this work numerical methods for the solution of two classes of structured generalized eigenvalue problems, $Ax=\lambda Bx$, are developed. Those classes are the palindromic ($B=A^T$) and the even ($A=A^T$, $B=-B^T$) eigenvalue problems.
The spectrum of these problems is not arbitrary, rather do eigenvalues occur in pairs.
We will construct methods for palindromic and even eigenvalue problems that are of cubic complexity and that are guaranteed to produce eigenvalues that are paired to working precision.
At the heart of both methods is a new URV-type matrix decomposition, that simultaneously transforms three matrices to skew triangular form, i.e., to a form that is triangular with respect to the Northeast-Southwest diagonal.
The algorithm to compute this URV decomposition uses several other methods to reduce a single square matrix to skew triangular form: the skew QR factorization and the skew QRQ$^T$ decomposition. Moreover, a method to compute the singular value decomposition of a complex, skew symmetric matrix is presented and used.
Being one of the key tools in conformation dynamics, the identification of
meta-stable states of Markov chains has been subject to extensive research in
recent years, especially when the Markov chains represent energy states of biomolecules. Some previous work on this topic involved the computation
of the eigenvalue cluster close to one, as well as the corresponding
eigenvectors and the stationary probability distribution of the associated stochastic
matrix. Later, since the eigenvalue cluster algorithm turned out to be non-robust, an optimisation approach was developed. As a possible less costly alternative, we present an SVD approach to identifying
meta-stable states of a stochastic matrix, where we only need
the second largest singular vector. We outline some theoretical background
and discuss the advantages of this strategy. Some simulated and real
numerical examples illustrate the effectiveness of the proposed algorithm.
We present a new extension of the well-known
Perron-Frobenius theorem to regular matrix pairs $(E,A)$.
The new extension is based on projector chains and is motivated from
the solution of positive differential-algebraic systems or descriptor
systems. We present several examples where the new condition holds, whereas conditions
in previous literature are not satisfied.
Perturbation of Purely Imaginary Eigenvalues of Hamiltonian Matrices under Structured Perturbations
(2007)
We discuss the perturbation theory for purely imaginary eigenvalues of Hamiltonian matrices under Hamiltonian and non-Hamiltonian perturbations. We
show that there is a substantial difference in the behavior under these perturbations. We also discuss the perturbation of real eigenvalues of real
skew-Hamiltonian matrices under structured perturbations and use these results to analyze the properties of the URV method of computing the
eigenvalues of Hamiltonian matrices.