C.S. Ballantine

Products of idempotent matrices

Abstract For some years it has been known that every singular square matrix over an arbitrary field F is a product of idempotent matrices over F. This paper quantifies that result to some extent. Main result: for every field F and every pair (n,k) of positive integers, an n×n matrix S over F is a product of k idempotent matrices over F iff rank(I − S)⩽k· nullity S. The proof of the “if” part involves only elementary matrix operations and may thus be regarded as constructive. Corollary: (for every field F and every positive integer n) each singular n×n matrix over F is a product of n idempotent matrices over F, and there is a singular n×n matrix over F which is not a product of n − 1 idempotent matrices.

Numerical range of a matrix: some effective criteria

Abstract Algorithms are presented which decide, for a given complex number w and a given complex n × n matrix S , whether w is in the numerical range W ( S ) of S , whether w is a boundary point of W ( S ), whether w is an extreme point of W ( S ), whether w is a bare point of W ( S ), and whether w is a vertex of W ( S ). Further algorithms decide whether W ( S ) intersects a given line (or a given ray), whether W ( S ) is included in a given open half plane (or a given closed half plane), and, for a given real number r , whether the numerical radius ρ s of S is > r , whether ρ s = r , and whether ρ s ≥ r . A simple effective criterion for H -stability is also given: a nonsingular H -semistable matrix S is H -stable iff the nullity of ( S + S ∗ ) S -1 ( S + S ∗ ) is twice the nullity of S + S ∗ . The computations involved in all these algorithms are elementary (rational operations, the max operation on pairs of real numbers, the degree of a nonzero polynomial, and the number of sign variations in the coefficients of a nonzero real polynomial), must be carried out exactly, and give exact (i.e., 100% reliable) results. Examples are worked out to illustrate the application of some of the algorithms.

Congruence and conjunctivity of matrices to their adjoints

Abstract Every square matrix over a field F is involutorily congruent over F to its transpose, and hence each such matrix is the product of a symmetric matrix and an involutory matrix over F . In the usual complex case every matrix which is conjunctive with its adjoint (=conjugate-transpose) is involutorily conjunctive with its adjoint and hence is the product of a hermitian matrix and an involutory matrix; furthermore every such matrix is conjunctive with a real matrix. These three conditions on a matrix, (1) being conjunctive with its adjoint, (2) being involutorily conjunctive with its adjoint, and (3) being conjunctive with a real matrix, are studied in the more general context of a field F with involution, and it is shown in general that (3) implies (2), that (2) implies (3) if char F ≠2 (a 2×2 counterexample exists for each F with char F =2), and that (1) does not in general imply (2) (a 2×2 counterexample in the complexification of the rational field is presented). The problem of deciding which matrices satisfy (2) is equivalent (even in this general context) to the problem of deciding which pairs of self-adjoint (“hermitian”) matrices are involutorily conjunctive. For the general 2×2 case, the three conditions are characterized in terms of norms.

Products of EP matrices

Abstract Necessary and sufficient conditions are given for a matrix to be a product of an EP r matrix by an EP s matrix. It is shown that a given square matrix is a product of more than two EP matrices of specified ranks (and hence nullities) if and only if its rank is less than or equal to the minimum of the given ranks and its nullity is less than or equal to the sum of the given nullities. It is also shown that given two EP matrices, the rank of their product is independent of the order of the factors.

A note on stable matrices

Abstract A new similarity-canonical form for stable matrices is given in the real and complex cases, and is used to derive new proofs for the (respective) real and complex cases of the Stein-Pfeffer theorem.

Simultaneous congruence of convex compact sets of Hermitian matrices with constant rank

Abstract Let S be a compact convex set of n × n hermitian matrices ( n ⩾ 2). Suppose every member of S is nonsingular and has exactly one negative eigenvalue. Let (e 1 ,…,e n ) be any ordered n -tuple from the set {- 1, 1}. One of our main results is that a nonsingular matrix X exists such that, for every A in S and every 1 ⩽ j ⩽ n , the ( j , j ) entry of X ∗ AX has sign e j . A similar result, with only negative e j allowed, is proved also for a compact convex set S of n × n hermitian matrices such that every member of S has the same rank and exactly one negative eigenvalue.