Stephen Pierce

Linear Preserver Problems

Linear preserver problems is an active research area in matrix and operator theory. These problems involve certain linear operators on spaces of matrices or operators. We give a general introduction to the subject in this article. In the first three sections, we discuss motivation, results, and problems. In the last three sections, we describe some techniques, outline a few proofs, and discuss some exceptional results. 1. EXAMPLES AND TYPICAL PROBLEMS. Let Mm,n be the set of m × n complex matrices, and let Mn = Mn,n. Suppose that M, N ∈ Mn satisfy det( MN ) = 1. Then the mapping φ : Mn → Mn given by

Extremal correlation matrices

Abstract Let R n denote the convex, compact set of all real n -by- n positive semidefinite matrices with main-diagonal entries equal to 1. We examine the extreme points of R n focusing mainly on their rank. the principal result is that R n contains extreme points of rank k if and only if k ( k +1)⩽2 n .

Linear maps on Hermitian matrices: the stabilizer of an inertia class, II

Let G(r s t) be the set of n-by-n Hermitian matrices with r positives negative and t zero eigenvaluesn = r + s + t. With the exception of the cases (i) r = n, (ii) s = n, and (iii) r = s t = 0 when n is even, we classify the nonsingulur linear maps T on Hermitian matrices for which T(G(r s t)) ⊆ G (r s t). Such a T is either a congruence or a congruence composed with transposition, with the additional possibility of composition with negation when r − sand t > 0. In cases (i) and (ii) above, there are definitely additional possible transformations, and a complete classification is a long standing unsolved problem. In case (iii) above, for n 4, we conjecture that the answer is congruence possibly composed with transposition and/or negation, but our methods do not cover this case. In two particular cases, (iv) r = n − 1s = 1t = 0 (n  3) and (v) r = s + 1t = 0. we show that the into assumption on T implies the nonsingularity of T, so that, in these cases, into alone implies that T is a congruence possibly c...

The ranks of extremal positive semidefinite matrices with given sparsity pattern

Let P be a symmetric set of ordered pairs of integers from 1 to n, and define

Chapter 9: miscellaneous preserver problems

M^ + (P)

Extremal bipartite matrices

to be the closed cone of all positive semidefinite Hermitian matrices whose

Linear preservers of the class of Hermitian matrices with balanced inertia

(i,j)

GENERALIZED MATRIX FUNCTION INEQUALITIES ON M-MATRICES

entry is zero whenever

Linear preservers of balanced nonsingular inertia classes

i \ne j

Linear maps on algebraic groups

and