A Decomposition Theorem for Matrices

Publisher Website
Google Scholar
Abstract
According to a classical theorem originally proved by L. Autonne (1; 3) in 1915, every m × n matrix of rank r with entries from the complex field can be decomposed as where U1 and U2 are unitary matrices of order m and n respectively and D is an m × n matrix having the form 1 where Δ is a non-singular diagonal matrix whose rank is r. If r = m, then the row of zero matrices of (1) does not actually appear. If r = n, then the column of zero matrices of (1) does not appear. The main purpose of this paper is to give a necessary and sufficient condition under which both U1 and U2 may be chosen to be real orthogonal matrices.

This publication has 1 reference indexed in Scilit: