Speaker
Description
Let $G, H \in \mathbb{C}^{n,n}$ be Hermitian and $S \in \mathbb{C}^{n,n}$ be a symmetric matrix. We consider the problem of maximizing the Rayleigh quotient of $G$ with respect to constraints involving symmetric matrix $S$ and Hermitian matrix $H$. More precisely, we compute
$$ \begin{align} m(G,H,S):=\sup\bigg\{ \frac{v^*Gv}{v^*v} :~v\in \mathbb{C}^{n} \setminus \{0\},\,v^TSv=0,\, v^*Hv=0 \bigg\},\tag{$\mathcal{P}$} \end{align} $$ where $T$ and $*$ denote respectively the transpose and the conjugate transpose of a matrix or a vector. Such problems occur in stability analysis of uncertain systems and in the eigenvalue perturbation theory of matrices and matrix polynomials [1, 2, 3]. A particular case of problem $(\mathcal{P})$ with only symmetric constraint (i.e., when $H=0$) is used to characterize the $\mu$-value of the matrix under skew-symmetric perturbations [3]. An explicit computable formula was obtained for $m(G,S)$ in [3, Theorem 6.3] and given by $$ m(G,S)=\inf_{t\in[0,\infty)} \lambda_2\left(\begin{bmatrix} G & t\overline S \\ t S & \overline G \end{bmatrix}\right), $$ where $\lambda_2(A)$ stands for the second largest eigenvalue of a Hermitian matrix $A$. The case where only Hermitian constraint (i.e., when $S=0$) was also considered and an explicit computable formula was obtained for $m(G,H)$ in [3, Theorem 6.2] and given by $$ m(G,H)=\inf_{t\in \mathbb{R}} \lambda_{max}(G+tH), $$ where $\lambda_{max}$ stands for the largest eigenvalue. However, the solution to the problem $(\mathcal{P})$ is still not known. ---------- [1] S. Bora, M. Karow, C. Mehl, P. Sharma. Structured eigenvalue backward errors of matrix pencils and polynomials with Hermitian and related structures. SIAM J. Matrix Anal. Appl., 35: 453--475 (2014). [2] J. Doyle. Analysis of feedback systems with structured uncertainties. IEE Proc. Part D, Control Theory Appl., 129: 242--250 (1982). [3] M. Karow. $\mu$-values and spectral value sets for linear perturbation classes defined by a scalar product. SIAM J. Matrix Anal. Appl., 32: 845--865 (2011).
