Multiplicative perturbation bounds for weighted polar decomposition
Let \(M\in\mathbb{C}^{m\times m}\) and \(N\in\mathbb{C}^{n\times n}\) be Hermitian positive definite, \(m\geq n\). Define on~\(\mathbb{C}^m\) the norm~\(\|\cdot\|_M\) by \(\|x\|_M^2= x^*Mx\), and on~\(\mathbb{C}^n\) the norm~\(\|\cdot\|_N\) similarly. The weighted conjugate transpose of \(B\in\mathbb{C}^{m\times n}\) is \(B^\sharp=N^{-1}B^*M\). If \(\|\cdot\|\) is a norm on~\(\mathbb{C}^{m\times n}\), the corresponding weighted norm of \(B\) is \(\|B\|_{(MN)}=\|M^\frac{1}{2}BN^{-\frac{1}{2}}\|\). The matrix~\(B\) is \((M,N)\) weighted partial isometric (\(MN\)-WPI) if \(\|Bx\|_M=\|x\|_N\) for all \(x\in R(B^\sharp)\), where \(R\) denotes the range. A matrix \(A\in\mathbb{C}^{m\times n}\) can be written as (1)~\(A=QH\), where \(Q\in\mathbb{C}^{m\times n}\) is \(MN\)-WPI and \(H\in\mathbb{C}^{n\times n}\) is such a matrix that \(NH\) is Hermitian nonnegative definite. Call (1) the \((M,N)\) weighted polar decomposition of~\(A\). Assume that \(R(Q^\sharp)=R(H)\); then it is unique. \smallskip A multiplicative perturbation of~\(A\) changes \(A\) to \(D_1AD_2\), where \(D_1\in\mathbb{C}^{m\times m}\) and \(D_2\in\mathbb{C}^{n\times n}\) are nonsingular. The authors find multiplicative perturbation bounds for \(Q\) and \(H\) under a weighted unitarily invariant norm, also for \(H\) in particular under the weighted Frobenius norm and weighted spectral norm, respectively. The bounds improve those by \textit{H. Li} et al. [Acta Math. Appl. Sin. 32, No. 5, 912--922 (2009; Zbl 1212.15025); Math. Inequal. Appl. 13, No. 3, 537--554 (2010; Zbl 1209.15023)].
scientific article; zbMATH DE number 6418688
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