A note on polar decomposition based Geršgorin-type sets
\textit{L. Smithies} and \textit{R. S. Varga} [Linear Algebra Appl. 417, No.~2--3, 370--380 (2006; Zbl 1101.15017)] developed a theoretical analysis for the construction of a Geršgorin-type set, \(\Gamma^{\text{SV}}(B)\), which contains the eigenvalues of an \(n \times n\) complex matrix \(B\) (\(n \leq \infty \)) using the set of all singular value decompositions (SVDs) of \(B\). A singular value decomposition of \(B\) is an expression of \(B\) as a product \(B=V \Sigma W^*\), where \(V\) and \(W^*\) are square unitary matrices and \(\Sigma := \text{Diag}(\sigma_1, \sigma_2, \dots, \sigma_n)\), with eigenvalues \(\sigma_1 \geq \sigma_2 \geq \dots \geq \sigma_n > 0\). A polar decomposition of a matrix \(B \in {\mathbb C}^{n \times n}\) is an expression of \(B\) as a product \(B=Q| B| \), where the factor \(| B| = \sqrt{(B*B)}\) is the unique absolute value of \(B\) and the factor \(Q\) is a partial isometry from the range of \(| B| \) onto the range of \(B\). In this paper, the Geršgorin-type set, \(\Gamma^{\text{PSV}}(B)\), constructed from the entries of the unique polar decomposition \(B=Q| B| \), is defined. It follows from this definition how a polar decomposition can be used to evaluate Geršgorin-type estimates of eigenvalues.
scientific article; zbMATH DE number 5362064