A min-max theorem for complex symmetric matrices (Q2575594)

This paper deals with the optimization of the form Re \(x^tTx\) to obtain the singular values of an \(n\times n\) (complex) symmetric matrix \(T\). The main result states that if \(\sigma_1\geq \cdots\geq \sigma_n\geq 0\) are the singular values of \(T\), then, for \(0\leq k< \frac n2\), \[ \min_{\text{ codim} \mathcal{V}=k}\,\max_{^{x\in\mathcal{V}}_{\| x\| =1}}\,\text{Re}\, x^tTx=\sigma_{2k+1} \] and \[ \min_{\dim \mathcal{V}=k}\max_{^{x\in\mathcal{V}}_{\| x\| =1}}\text{Re}\, x^tTx=0\, , \] where \(\mathcal{V}\) runs over the complex subspaces of \(\mathbb{C}^n\), such that codim \(\mathcal{V}=k\) and \(\dim\mathcal{V}=k\), respectively.