If \(A\) is a complex \(n \times n\) matrix then its departure from normality is defined as \(\text{dep}_ F (A) : = (\| A \|^ 2_ F - \| \text{diag} (\lambda_ 1, \dots, \lambda_ n) \|^ 2_ F )^{1/2} \), where \(\| \cdot \|_ F\) is the Frobenius norm and the \(\lambda_ i\)’s are the eigenvalues of \(A\). With \(\widetilde A : = A - {1 \over n} \text{tr} (A)I\) the author establishes that \(\text{dep}_ FA \geq (\| \widetilde A \|^ 2_ F - (\| \widetilde A \|^ 4_ F - {1 \over 2} \| A^ H A - AA^ H \|^ 2_ F)^{1/2})^{1/2}\), \(\text{dep}_ FA \leq (\| \widetilde A \|^ 2_ F - | \text{tr} (\widetilde A^ 2) |)^{1/2}\).
These bounds improve results by P. J. Eberlein [Amer. Math. Monthly 72, 995-996 (1965; Zbl 0142.003)], L. Elsner and M. H. C. Paardekooper [Linear Algebra Appl. 92, 107-124 (1987; Zbl 0621.15014)], G. Loizou [J. Assoc. Comp. Mach. 16, 580-584 (1969; Zbl 0188.079)], and R. Kress, H. L. de Vries and R. Wegmann [Linear Algebra Appl. 8, 109-120 (1974; Zbl 0273.15018)]. The upper bound does not necessarily reduce to zero for normal matrices. Moreover, new bounds for \(\| \text{diag} (\lambda_ 1, \dots, \lambda_ n) \|^ 2_ F\), the spectral radius and the spread of \(A\) are given.
For the entire collection see [Zbl 0809.00014].