A bound for the condition of a hyperbolic eigenvector matrix
Let \(H\) be a Hermitian positive definite matrix and \(J= \text{diag}(\pm 1)\). The starting point of the paper is the hyperbolic eigenvalue problem \(Hx=\lambda Jx\). It is known that there is always a matrix \(X\) such that \(X^*HX =\Lambda\), and \(X^*JX=J\), where \(\Lambda\) is a diagonal positive definite matrix. The matrix \(X\) is called a hyper-exchange matrix with respect to the signature matrix \(J\) and occurs in a variety of linear algebra problems. If \(\|\cdot\|\) denotes the spectral matrix norm, the spectral condition \(\kappa(A)\) of a nonsingular matrix \(A\) is given by \(\kappa (A)= \|A\|\cdot\|A^{-1}\|\). It is known that all the matrices \(X\) above have the same condition \(\kappa(X)= \|X\|^2\). A main result of the present paper is that \(\kappa(X) \leq\sqrt {\min \kappa(D^*HD)}\), where the minimum is over all nonsingular matrices \(D\) that commute with \(J\). It is also shown for which matrices \(D\) the minimum is attained and for which matrices \(H\) the bound itself is attained. Some applications are given to other related systems.
scientific article; zbMATH DE number 1331377
scientific article