Crawford numbers of powers of a matrix

or an n-by-n matrix A, its Crawford number c(A) (resp., generalized Crawford number C(A)) is, by definition, the distance from the origin to its numerical range W(A) (resp., the boundary of its numerical range ∂W(A)). It is shown that if A has eigenvalues λ₁,⋯, λ_n arranged so that |λ ₁|≥⋯≥|λ_n|, then lim _kc(A^k)^1/k (resp., lim_kC(A ^k)^1/k) equals 0 or |λn| (resp., |λj| for some j, 1≤j≤n). For a normal A, more can be said, namely, lim _kc(A(A^k)^1/k=|λ_n| (resp., lim_kC((A^k)^1/k=|λj| for some j, 3≤j≤n). In these cases, the above possible values can all be assumed by some A.