Distances from a Hermitian Pair to Diagonalizable and Nondiagonalizable Hermitian Pairs
Chi-Kwong Li, Roy Mathias · SIAM Journal on Matrix Analysis and Applications · 2006
Let $W(T)$ and $r(T)$ denote the numerical range and numerical radius of an $n \times n$ complex matrix T. Let $H_n^2$ denote the space of pairs of $n \times n$ Hermitian matrices. Define a norm on $H_n^2$ by $\|(X, Y)\| = r(X + iY)$. Take $(A,B) \in H_n^2$. It is shown that if $0 \in W(A+iB)$, then $\inf\{ |\mu|: \mu otin W(A+iB)\}$ is an upper bound on the distance to the nearest pair that is simultaneously diagonalizable by congruence. If $0 otin W(A+iB)$, then $\min \{ |\mu|: \mu \in W(A+iB)\}$, which is the Crawford number of the pair $(A, B)$, is equal to the distance to the nearest pair that is not simultaneously diagonalizable by congruence. The results are similar when the numerical radius is replaced by the spectral norm.