Inclusion sets for the singular values of a square matrix

The paper presents a general approach to deriving various inclusion sets for the singular values of a matrix $A=(a_{ij})\in\mathbb C^{n\times n}$. The key to the approach is the following result: If $\sigma$ is a singular value of $A$, then a certain matrix $C(\sigma,A)$ of order $2n$, whose diagonal entries are $\sigma^2-|a_{ii}|^2$, $i=1,\dots,n$, is singular. Based on this result, we use known diagonal-dominance type nonsingularity conditions to obtain inclusion sets for the singular values of $A$. Scaled versions of the inclusion sets, allowing one, in particular, to obtain Ky Fan type results for the singular values, are derived by passing to the conjugated matrix $D^{-1}C(\sigma,A)D$, where $D$ is a positive-definite diagonal matrix. Bibl. – 16 titles.