Singular Values, Eigenvalues and Diagonal Elements Of The Commutator Of 2 x 2 Rank One Matrices

Abstract

The region of singular values of the commutator XY − Y X for 2 × 2 rank one complex matrices X and Y is determined. This answers in affirmative a conjecture raised in [D. Wenzel. A strange phenomenon for the singular values of commutators with rank one matrices. Electron. J. Linear Algebra, 30:649–669, 2015.] when 2 × 2 matrices are concerned. The approach and proofs also lead to a complete relation between the singular values, eigenvalues and diagonal elements of the commutator under consideration.