Leiba Rodman

Spectral analysis of matrix polynomials— I. canonical forms and divisors

Abstract The Jordan normal form for complex matrices is extended to admit “canonical triples” of matrices for monic matrix polynomials and to “standard triples” of operators for monic operator polynomials on finite dimensional linear spaces. These ideas lead to the formulation of canonical, or standard, forms for such polynomials. The inverse problem is also investigated: When do triples of matrices (operators) determine a monic matrix (operator) polynomial for which the triple is canonical (standard)? The canonical and standard forms are very well suited to the study of division and multiplication processes. This is carried out in detail with special emphasis on the question of when matrix (or operator) polynomials have nontrivial divisors which are polynomials of the same kind.

Stability of invariant maximal semidefinite subspaces. I

Abstract We study stability in classes of subspaces which are invariant under a self-adjoint matrix in an indefinite inner product, and have various maximality and semidefiniteness properties with respect to this indefinite inner product. Descriptions of all subspaces in such a class for which these properties are stable in one or another way are obtained.

Minimal Divisors of Rational Matrix Functions with Prescribed Zero and Pole Structure

Necessary and sufficient conditions are given in order that a rational matrix function is a minimal divisor of another one. These conditions are expressed in terms of zero and pole structure of the given functions. In connection with this a description is obtained of all rational matrix functions with prescribed zero and pole data.

Solutions of the Continuous and Discrete Time Algebraic Riccati Equations: A Review

This review is concerned with two algebraic Riccati equations. The first is a quadratic matrix equation for an unknown n × n matrix X of the form n n

Spectral analysis of matrix polynomials— II. The resolvent form and spectral divisors

XDX + XA + A*X - C = 0,

Analytic operator functions with compact spectrum. I. Spectral nodes, linearization and equivalence

n n(2.1) n nwhere A, D, C are n × n complex matrices with C and D hermitian. Further hypotheses are imposed as required, although Section 2.3 contains some discussion of more general non-symmetric quadratic equations. The second equation has the fractional form n n

Analytic operator functions with compact spectrum. II. Spectral pairs and factorization

X = A*XA + Q - (C + B*XA)*{(R + B*XB)^{ - 1}}(C + B*XA),