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SIAM J. on Matrix Analysis and Applications

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1988

Volume 9, Issue 4, pp. 455-593


Classifications of Nonnegative Matrices Using Diagonal Equivalence

Daniel Hershkowitz, Uriel G. Rothblum, and Hans Schneider

SIAM. J. Matrix Anal. & Appl. 9, pp. 455-460 (6 pages)

Online Publication Date: July 17, 2006

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This article studies matrices $A$ that are positively diagonally equivalent to matrices that, for given positive vectors $u,v,r,$ and $c$, map $u$ into $r$, and where $A^T $ map $v$ into $c$. The problem is reduced to scaling a matrix for given row sums and column sums, and applying known results for the latter. Further classifications that use these results are investigated.

Linear Preservers of the Class of Hermitian Matrices with Balanced Inertia

Stephen Pierce and Leiba Rodman

SIAM. J. Matrix Anal. & Appl. 9, pp. 461-472 (12 pages) | Cited 2 times

Online Publication Date: July 17, 2006

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Let $H ( n )$ the $n^2 $-dimensional real vector space of Hermitian matrices. Assume $n$ is even and greater than or equal to 4. Let $T$ be an invertible linear transformation on $H ( n )$ that maps the class of invertible, balanced inertia (signature zero) Hermitian matrices into itself. Then for some real number $c \ne 0$, and an invertible matrix $S,T ( A ) = cS^* AS$ or $T ( A ) = cS^* A^T S$, for all $A \in H( n )$. $T$ is also classified in the case where $n=2$.

On Minimizing the Special Radius of a Nonsymmetric Matrix Function: Optimality Conditions and Duality Theory

Michael L. Overton and Robert S. Womersley

SIAM. J. Matrix Anal. & Appl. 9, pp. 473-498 (26 pages) | Cited 4 times

Online Publication Date: July 17, 2006

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Let $A( x )$ be a nonsymmetric real matrix affine function of a real parameter vector $x \in \mathcal{R}^m $, and let $\rho ( x )$ be the spectral radius of $A( x )$. The article addresses the following question: Given $x_0 \in \mathcal{R}^m $, is $\rho ( x )$ minimized locally at $x_0 $, and, if not, is it possible to find a descent direction for $\rho ( x )$ from $x_0 $? If any of the eigenvalues of $A( x_0 )$ that achieve the maximum modulus $\rho ( x_0 )$ are multiple, this question is not trivial to answer, since the eigenvalues are not differentiable at points where they coalesce. In the symmetric case, $A( x ) = A( x )^T $ for all $x,\rho ( x )$ is convex, and the question was resolved recently by Overton following work by Fletcher and using Rockafellar’s theory of subgradients. In the nonsymmetric case $\rho ( x )$ is neither convex nor Lipschitz, and neither the theory of subgradients nor Clarke’s theory of generalized gradients is applicable. A new necessary and sufficient condition is given for $\rho ( x )$ to have a first-order local minimum at $x_0 $, assuming that all multiple eigenvalues of $A( x_0 )$ that achieve the maximum modulus are nondefective. The optimality condition is computationally verifiable and involves computing “dual matrices.” If the condition does not hold, the dual matrices provide information that leads to the generation of a descent direction. The result can be extended to the case where $\rho ( x )$ is replaced by the maximum real part of the eigenvalues of $A( x )$. The authors use the eigenvalue perturbation theory of Rellich and Kato, which provides expressions for directional derivatives of $\rho ( x )$. They also derive formulas for the codimension of manifolds on which certain eigenvalue structures of $A( x )$ are maintained; these are due to Von Neumann and Wigner and to Arnold. Finally, they discuss the much more difficult question of resolving optimality when $A( x_0 )$ has a defective multiple eigenvalue achieving the maximum modulus $\rho ( x_0 )$.

The Periodic Lyapunov Equation

Paolo Bolzern and Patrizio Colaneri

SIAM. J. Matrix Anal. & Appl. 9, pp. 499-512 (14 pages) | Cited 12 times

Online Publication Date: July 17, 2006

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This paper presents an overview of the periodic Lyapunov equation, both in discrete time and in continuous time. Together with some selected results that have recently appeared in the literature, the paper provides necessary and sufficient conditions for the existence and uniqueness of periodic solutions.

Block-Sequential Algorithms for Set-Theoretic Estimation

Ronald K. Pearson

SIAM. J. Matrix Anal. & Appl. 9, pp. 513-527 (15 pages) | Cited 6 times

Online Publication Date: July 17, 2006

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A new algorithm is proposed for solving the set-theoretic parameter estimation problem. In contrast to point estimation strategies like least squares or maximum likelihood, the set-theoretic parameter estimation problem imposes bounds on model errors and seeks the resulting bounds imposed on the free model parameters. The exact solution to this problem is a convex polytope in the parameter space with too many vertices for an exact solution to be practical. Thus, the standard solution approach is to seek an outer bounding set that is more easily parameterized. This paper describes a computational approach that interpolates in estimation efficiency and computational effort between two extreme cases described by other authors. Two simple numerical examples are included.

Homotopy Method for General $\lambda $-Matrix Problems

Moody T. Chu, T. Y. Li, and Tim Sauer

SIAM. J. Matrix Anal. & Appl. 9, pp. 528-536 (9 pages) | Cited 11 times

Online Publication Date: July 17, 2006

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This paper describes a homotopy method used to solve the $k$th-degree $\lambda $-matrix problem $( A_k \lambda ^k + A_{k - 1} \lambda ^{k - 1} + \cdots + A_1 \lambda + A_0 )x = 0$. A special homotopy equation is constructed for the case where all coefficients are general $n\times n$ complex matrices. Smooth curves connecting trivial solutions to desired eigenpairs are shown to exist. The homotopy equations maintain the nonzero structure of the underlying matrices (if there is any) and the curves correspond only to different initial values of the same ordinary differential equation. Therefore, the method might be used to find all isolated eigenpairs for large-scale $\lambda $-matrix problems on single-instruction multiple data (SIMD) machines.

Cycle Lengths in $A^k b$

Charles M. Grinstead

SIAM. J. Matrix Anal. & Appl. 9, pp. 537-542 (6 pages)

Online Publication Date: July 17, 2006

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Let $A$ be a nonnegative, $n \times n$ matrix, and let $b$ be a nonnegative, $n \times n$ vector. Let $S$ be the sequence $\{ A^k b \},k = 0,1,2, \cdots $. Define $m( A,b )$ to be the length of the cycle of zero-nonzero patterns into which $S$ eventually falls. Define $m( A )$ to be the maximum, over all nonnegative $b$ of $m( A,b )$. Finally, define $m( n )$ to be the maximum, over all nonnegative, $n \times n$ matrices $A$ of $m( A )$. This paper shows given $A$ and $b$, that $m( A,b )$ is a divisor of a certain number, which is determined by the structure of $A$ and $b$. It is also shown that $\log m ( n ) \sim ( n\log n )^{1/ 2} $.

Eigenvalues and Condition Numbers of Random Matrices

Alan Edelman

SIAM. J. Matrix Anal. & Appl. 9, pp. 543-560 (18 pages) | Cited 78 times

Online Publication Date: July 17, 2006

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Given a random matrix, what condition number should be expected? This paper presents a proof that for real or complex $n \times n$ matrices with elements from a standard normal distribution, the expected value of the log of the 2-norm condition number is asymptotic to $\log n$ as $n \to \infty$. In fact, it is roughly $\log n + 1.537$ for real matrices and $\log n + 0.982$ for complex matrices as $n \to \infty$. The paper discusses how the distributions of the condition numbers behave for large $n$ for real or complex and square or rectangular matrices. The exact distributions of the condition numbers of $2 \times n$ matrices are also given.
Intimately related to this problem is the distribution of the eigenvalues of Wishart matrices. This paper studies in depth the largest and smallest eigenvalues, giving exact distributions in some cases. It also describes the behavior of all the eigenvalues, giving an exact formula for the expected characteristic polynomial.

A Note on the Newoton Iteration for the Algebraic Eigenvalue Problem

Maria Célia Santos

SIAM. J. Matrix Anal. & Appl. 9, pp. 561-569 (9 pages)

Online Publication Date: July 17, 2006

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This paper considers the Newton iteration for the algebraic eigenvalue problem $( \lambda I - A )x = 0,\Phi ( x ) = 1$, where $\Phi $ is a convex forming function that is not necessarily differentiable. The role usually played by the Fréchet or Gateaux derivatives will be performed by a choice of subgradients of $\Phi $. Under very mild conditions on $\Phi $, the local and $Q$-superlinear convergence of this extended Newton iteration are proved. The stability of the process is also investigated.

Linear Matrix Equations Controllability and Observability, and the Rank of Solutions

Harald K. Wimmer

SIAM. J. Matrix Anal. & Appl. 9, pp. 570-578 (9 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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The equation \[ (*) \qquad \sum_{i,k} f_{ik} A^i XB^k = C \] is studied. The controllability matrix of $( A,C )$ and the observability matrix of $( B,C )$ yield bounds for the rank of $X$. If the solution $X$ is unique it can be expressed in the form \[ X = \sum_{i,k} h_{ik} A^i CB^k . \] The coefficients $h_{ik} $ are determined by an auxiliary equation of type $( * )$, where the right-hand side is a rank one matrix.

Five-Diagonal Toeplitz Determinants an Their Relation to Chebyshev Polynomials

Robert B. Marr and George H. Vineyard

SIAM. J. Matrix Anal. & Appl. 9, pp. 579-586 (8 pages) | Cited 2 times

Online Publication Date: July 17, 2006

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A five-diagonal Toeplitz (5DT) determinant is defined as having zeros everywhere except in its five principal diagonals, with each principal diagonal having the same element in all positions. Thus the determinant depends on five arbitrary parameters in addition to its order. The general 5DT determinant of order $n$ is shown to be given by a simple closed expression involving Chebyshev polynomials of the second kind of order $n + 1$. An explicit generating function for the determinants is also derived such that the $n$th coefficient of a power series expansion of the function is the $n$th-order five-diagonal Toeplitz determinant.

A Sharp Bound for Products of Hyperbolic Plane Rotations

C. T. Pan and Kermit Sigmon

SIAM. J. Matrix Anal. & Appl. 9, pp. 587-593 (7 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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An algorithm for downdating a least squares problem using hyperbolic plane rotations has recently been presented and analyzed by Alexander, Pan, and Plemmons. Their analysis of the numerical stability of the algorithm rests on the existence of a tight bound on the product of the norms of a certain collection of hyperbolic rotations. The main result of this paper, which was obtained in conjunction with that work, establishes the required tight bound. The inequality established may be of interest in its own right.
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