Condition numbers of matrices with given spectrum (Q2009426)

This paper surveys recent strategies to estimate the condition number \(\operatorname{CN}(T)=||T||||T^{-1}||\) of complex \(n\times n\) matrices \(T\) with a given spectrum. More precisely, the authors present a proof of the fact that if \(T\) acts on the Hilbert space \(C_n\), then the supremum of \(\operatorname{CN}(T)\) over all contractions \(T\) with smallest eigenvalues of modulus \(r > 0\), is equal to \(1/r_n\). This is achieved using an analytic Toeplitz matrix. The same question is posed for \(n\)-dimensional Banach spaces. These strategies provide explicit and constructive solutions to the so-called Halmos and Schäffer problems and are also shown to be effective in a closely related situation, namely considering Kreiss matrices instead of contractions.