A constructive version of the Boyle–Handelman theorem on the spectra of nonnegative matrices

AbstractA constructive version of the celebrated Boyle–Handelman theorem on the non-zero spectra of nonnegative matrices is presented

Irredundant generating sets for matrix algebras

AbstractWe prove the following Main Theorem: Let S be an irredundant generating set for the algebra Mn(F) of n × n matrices over a field F. Then the cardinality of S is at most 3 if n = 2, and at most 2n − 2 if n > 2

Simultaneous quasidiagonalization of complex matrices

AbstractLet be the complex algebra generated by a pair of n × n Hermitian matrices A, B. A recent result of Watters states that A, B are simultaneously unitarily quasidiagonalizable [i.e., A and B are simultaneously unitarily similar to direct sums C1⊕…⊕Ct,D1⊕…⊕Dt for some t, where Ci, Di are ki × ki and ki⩽2(1⩽i⩽t)] if and only if [p(A, B), A]2 and [p(A, B), B]2 belong to the center of for all polynomials p(x, y) in the noncommuting variables x, y. In this paper, we obtain a finite set of conditions which works. In particular we show that if A, B are positive semidefinite, then A, B are simultaneously quasidiagonalizable if (and only if) [A, B]2, [A2, B]2 and [A, B2]2 commute with A, B

Factorizations of integer matrices as products of idempotents and nilpotents

AbstractIt is proved that for n ⩾ 3, every n ×n matrix with integer entries and determinant zero is the product of 36n + 217 idempotent matrices with integer entries

Two-generated commutative matrix subalgebras

AbstractLet F be a field, and let Mn(F) be the algebra of n×n matrices over F. Let A, B ∈ Mn(F) with AB = BA, and let A be the algebra generated by A, B over F.A theorem of Gerstenhaber [Ann. Math. 73:324–348 (1991)] states that the dimension of A is at most n. Gerstenhaber's proof uses the methods of algebraic geometry. In this paper, we obtain a purely matrix-theoretic proof of the result. We also examine the case when equality occurs. The case where F is algebraically closed and A is indecomposable (under similarity) holds the key to the general situation, and in the indecomposable case, we obtain a Cayley-Hamilton-like theorem expressing Bk as a polynomial in I,B,…, Bk−1 with coefficients in F[A], where k denotes the number of blocks in the Jordan form of A. If all Jordan blocks of A have the same size, we obtain a nonderogatory-like condition on B which is equivalent to dimFA = n. We also show that in this case dimFA = n is equivalent to the maximality of A as a commutative subalgebra of Mn(F)

Functions preserving nonnegativity of matrices

The main goal of this work is to determine which entire functions preserve nonnegativity of matrices of a fixed order $n$ -- i.e., to characterize entire functions $f$ with the property that $f(A)$ is entrywise nonnegative for every entrywise nonnegative matrix $A$ of size $n\times n$. Towards this goal, we present a complete characterization of functions preserving nonnegativity of (block) upper-triangular matrices and those preserving nonnegativity of circulant matrices. We also derive necessary conditions and sufficient conditions for entire functions that preserve nonnegativity of symmetric matrices. We also show that some of these latter conditions characterize the even or odd functions that preserve nonnegativity of symmetric matrices.Comment: 20 pages; expanded and corrected to reflect referees' remarks; to appear in SIAM J. Matrix Anal. App

Knowledge-based monitoring of the pointing control system on the Hubble space telescope

A knowledge-based system for the real time monitoring of telemetry data from the Pointing and Control System (PCS) of the Hubble Space Telescope (HST) that enables the retention of design expertise throughout the three decade project lifespan by means other than personnel and documentation is described. The system will monitor performance, vehicle status, success or failure of various maneuvers, and in some cases diagnose problems and recommend corrective actions using a knowledge base built using mission scenarios and the more than 4,500 telemetry monitors from the HST