Finding The Closest Persymmetric And Skew-Symmetric Matrices.

Abstract

The class of Procrustes problems has many application in the biological, physical
and social sciences just as in the investigation of elastic structures. The problem
consists of solving a constrained linear least squares problem defined on a set of the space
of matrices . The different problems are obtained varying the structure of the matrices
belonging to the feasible set. Higham has solved the orthogonal, the symmetric and the
positive definite cases. Raydán has studied the rectangular case and minimizing on a feasible
set which is an intersection of convex sets of matrices. The method used is based on
the alternate projection method. The Toeplitz cases has been analyzed by these authors.
In this contribution, the theory and algorithm developed by Higham for the symmetric
Procrustes problem are extended to the persymmetric and skew-symmetric cases. The
singular value decomposition is used to analyze the problems and to characterized their
solutions.
Numerical difficulties are discussed and illustrated by examples.