Antieigenvalues of doubly stochastic matrices (Q2904184)

Let \(A\) be an operator on a complex Hilbert space \(H\) and \(\theta \in \mathbb{R}.\) Then the \(\theta\)-antieigenvalue \(\mu_{\theta }(A)\) is defined to be NEWLINE\[NEWLINE\inf _{Af \neq \theta}\frac {\cos \theta Re<Af,f>+\sin \theta <Af,f>}{\|Af\|~\|f\|}.NEWLINE\]NEWLINE The vectors \(f\) (if exist) for which \(\mu_{\theta }(A)\) is obtained are called \(\theta\)-antieigenvectors. The authors compute \(\theta\)-antieigenvectors for \(2\times 2\) complex matrices. They also compute the antieigenvalue of a \(3\times 3\) doubly stochastic matrix.