Structured Mapping Problems for Matrices Associated with Scalar Products. Part I: Lie and Jordan Algebras

Abstract

Abstract. Given a class of structured matrices S, we identify pairs of vectors x, b for which there exists a matrix A ∈ S such that Ax = b, and also characterize the set of all matrices A ∈ S mapping x to b. The structured classes we consider are the Lie and Jordan algebras associated with orthosymmetric scalar products. These include (skew-)symmetric, (skew-)Hamiltonian, pseudo (skew-)Hermitian, persymmetric and perskew-symmetric matrices. Structured mappings with extremal properties are also investigated. In particular, structured mappings of minimal rank are identified and shown to be unique when rank-1 is achieved. The structured mapping of minimal Frobenius norm is always unique and explicit formulas for it and its norm are obtained. Finally the set of all structured mappings of minimal 2-norm is characterized. Our results generalize and unify existing work, answer a number of open questions, and provide useful tools for structured backward error investigations.

A structure-preserving QR factorization for centrosymmetric real matrices

“…This problem was completely solved in [10] for orthosymmetric scalar products. General structured mapping problems were treated in [7]. In [5] an algorithm for obtaining eigenvalues of matrices that are symmetric and persymmetric (which are a strict subset of centrosymmetric matrices) by applying structured similarities and reducing a given matrix to an "X-form".…”

A second Wedderburn-type theorem for some classes of linearly structured matrices

“…Fix K ∈ {R, C} and denote M n (K) the set of (n × n)-matrices with entries in K. Write the transpose of A ∈ M n (K) by A T and the conjugate transpose by A H . If we fix a non-singular symmetric or Hermitian M ∈ M n (K), then we get the following Lie and where P = T when K = R or P = H when K = C. This provides us with a general framework to study important classes of structured matrices like the ones of Hamiltonian, skew-Hamiltonian, symmetric, skew-symmetric, pseudosymmetric, persymmetric, Hermitian, skew-Hermitian, pseudo-Hermitian, pseudo-skew-Hermitian matrices and so on (see [5,6]). …”

Structured Eigenvalue Backward Errors of Matrix Pencils and Polynomials with Hermitian and Related Structures

“…These mapping problems can be condensed into the following general Hermitian mapping problem: Under which conditions on vectors x, y ∈ C n does there exist a Hermitian matrix H ∈ C n×n satisfying Hx = y? The answer to this problem is well known; see, e.g., [27] where solutions that are minimal with respect to the spectral or Frobenius norm are also characterized. We also refer to [19] and [32] for the more general problem of the existence of a Hermitian H ∈ C n×n such that HX = Y for two matrices X, Y ∈ C n×m .…”