Condition numbers of matrices with given spectrum

This note surveys recent strategies to estimate the condition number CN(T)=‖ T‖·‖ T^-1‖ of complex n× n matrices T with given spectrum. More precisely, we present a proof of the fact that if T acts on the Hilbert space ℂ^n , then the supremum of CN ( T ) over all contractions T with smallest eigenvalues of modulus r>0 , is equal to 1/r^n , and is achieved by an analytic Toeplitz matrix. The same question is treated for n -dimensional Banach spaces. These strategies provide with explicit and constructive solutions to the so-called Halmos and Schäffer’s problems, and are also shown to be effective in a closely related situation, namely considering Kreiss matrices instead of contractions.