Distances from a Hermitian Pair to Diagonalizable and Nondiagonalizable Hermitian Pairs

Denote by W (T ), r(T ) and ‖T‖ the numerical range, the numerical radius and the spectral norm of a complex matrix T . Let (A,B) be a pair of Hermitian matrices. It is shown that if 0 ∈ W (A+ iB) then d(A,B) = inf{|μ| : μ / ∈ W (A+ iB)} is an upper bound for inf{r(E + iF ) : (A+ E) + i(B + F ) is diagonalizable by congruence}; if 0 / ∈ W (A+ iB) then the Crawford number c(A,B) = min{|μ| : μ ∈ W (A+ iB)} is equal to min{r(E + iF ) : (A+ E) + i(B + F ) is not diagonalizable by congruence}, which in turn is equal to inf{‖E + iF‖ : (A+ E) + i(B + F ) is not diagonalizable by congruence}. The infimum is not always attained. AMS Subject Classifications: 15A60, 15A18.