Stability of self-adjoint square roots and polar decompositions in indefinite scalar product spaces
Let \(F\) be either the real or complex numbers and let \(H\) be an invertible matrix that is either real symmetric or complex Hermitian as appropriate. The usual scalar product \(\langle x,y\rangle\) on \(F^n\) can then be modified by \(H\) to give a perhaps indefinite scalar product \([x,y]=\langle Hx,y \rangle\). The \(H\)-adjoint \(A^H\) of an \(n\times n\) matrix \(A\) is defined by \([Ax, y]= [x,A^Hy]\) and an \(n\times n\) matrix \(U\) is \(H\)-unitary if \([Ux,Uy] =[x,y]\). An \(H\)-polar decomposition of a square matrix \(X\) is one of the form \(X=UA\) with \(U\) \(H\)-polar and \(A\) \(H\)-self-adjoint. The authors are concerned with the existence of stable \(H\)-polar decompositions, that is, decompositions that depend continuously on \(X\) for small perturbations, when the scalar product is genuinely indefinite. In the complex case, such a decomposition occurs if and only if the eigenvalues of \(X^HX\) are positive. In the real case, a further requirement is needed for necessity. The authors also consider related questions concerning matrices that can be written as \(X^HX\), and those matrices \(Y\) having an \(H\)-self-adjoint square root. A characterization is given of such \(Y\) whose square root depends continuously on \(Y\).
scientific article; zbMATH DE number 1419861
scientific article