Inclusion sets for the singular values of a square matrix

The paper presents a general approach to deriving inclusion sets for the singular values of a matrix A = (a_ij) ∈ ℂ ^n×n. The key to the approach is the following result: If σ is a singular value of A, then a certain matrix C(σ, A) of order 2n, whose diagonal entries are σ² − | a_ii|², i = 1, …, 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⁻¹C(σ, A)D, where D is a positive-definite diagonal matrix. Bibliography: 16 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 52–77.