The ψ S polar decomposition

• Published in: Linear algebra and its applications 2009-09, Vol.431 (8), p.1249-1256
• Main Authors: Horn, Roger A., Merino, Dennis I., Paras, Agnes T.
• Format: Article
• Language: English
• Subjects: [formula omitted] antiorthogonal matrices; [formula omitted] orthogonal matrices; [formula omitted] polar decomposition; [formula omitted] symmetric matrices; Algebra; Coninvolutory matrices; Exact sciences and technology; Linear and multilinear algebra, matrix theory; Mathematics; Sciences and techniques of general use; Skew-coninvolutory matrices
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2009.04.021

Let S ∈ M n be nonsingular. We set ψ S ( A ) = S - 1 A ¯ - 1 S for all nonsingular A ∈ M n ; a matrix A is called ψ S symmetric if ψ S ( A ) = A , it is called ψ S orthogonal if ψ S ( A ) = A - 1 , and it is called ψ S antiorthogonal if ψ S ( A ) = - A - 1 . We show that the following are equivalent: (1) A is ψ S symmetric, (2) there exists a ψ S antiorthogonal Z ∈ M n such that A = e Z , (3) there exists a ψ S orthogonal X ∈ M n such that A = e iX , and (4) there exists a ψ S symmetric B ∈ M n such that A = B 2 . When S is coninvolutory ( S S ¯ = I ) or skew-coninvolutory ( S S ¯ = - I ) , we show that every nonsingular matrix has a ψ S polar decomposition, that is, every nonsingular matrix may be written as A = RE , where R is ψ S orthogonal and E is ψ S symmetric. If A is possibly singular, we define A to be ψ S orthogonal if S - 1 A ¯ S = A and determine which singular matrices have a ψ S polar decomposition.