A note on the rank-subtractivity ordering
Among known partial orderings of the set \(\mathbb{C}_{m,n}\) of all complex \(m\times n\) matrices are \(A{^{rs}\leq} B\) (``rank-subtractivity'') defined by \(A{^{rs}\leq} B\Leftrightarrow \text{rk}(B-A)=\text{rk}(B)-\text{rk}(A)\), and \(A{^\sigma\leq}B\) (``singular values'') defined by \(A{^\sigma\leq}B\Leftrightarrow A{^{rs}\leq}B\) and \(\sigma(A)\subseteq\sigma(B)\), where \(\sigma(X)\) is the set of singular values of \(X\). The author introduces a new partial ordering \(A{^{\sigma_1}\leq}B\) intermediate between these two: \(A{^{\sigma_1}\leq}B\Leftrightarrow A{^{rs}\leq}B\) and \(\sigma_1(A) \leq\sigma_1(B)\), where \(\sigma_1(X)\) is the largest singular value of \(X\). He investigates its formal properties and its relations to other partial orderings. An application to multivariate analysis is given.
scientific article; zbMATH DE number 1331297
scientific article