Spectral variation bounds for diagonalisable matrices (Q1364855)

Let \(A\) and \(B\) be \(n\times n\) complex diagonalizable matrices with real eigenvalues \(\alpha_1\geq \cdots \geq \alpha_n\) and \(\beta_1\geq \cdots \geq\beta_n\) respectively, and suppose \(SAS^{-1}\) and \(TBT^{-1}\) are diagonal. Let \(c(X)= \| X\| \| X^{-1} \|\) where \(\|\cdot \|\) is the operator bound norm. Let \(|\| \cdot| \|\) denote any unitarily invariant norm. In the earlier paper by \textit{R. Bhatia}, \textit{C. Davis} and \textit{F. Kittaneh} [Aequationes Math. 41, No. 1, 70-78 (1991; Zbl 0752.47005)] it was proved that \[ |\| \text{diag} (\alpha_1- \beta_1, \dots, \alpha_n- \beta_n) |\|\leq c(S)c(T) | \| A-B |\|. \] Here it is proved that \[ |\| A-B |\|\leq c(S)c(T) |\| \text{diag} (\alpha_1- \beta_n, \dots, \alpha_n- \beta_1)| \|. \] These generalize inequalities known true for Hermitian matrices. Related inequalities are discussed and an earlier conjecture is shown to be false.