Generalized frame operator distance problems

RIOS, NOELIA B.; PEDRO MASSEY; DEMETRIO STOJANOFF

Rio de Janeiro

Congreso; 22nd conference of the international linear algebra society; 2019

International linear algebra society

Lidskii's additive inequality - in terms of majorization of eigenvalues - can be interpreted as a description of a family of global minimizersof functions $\Phi_{S,\,N}=\Phi:\mathcal O_\mu\rightarrow \mathbb R_{\geq 0}$ of the form $\Phi(G)=N(S-G)$, where$N$ is a unitarily invariant norm (uin) in $\mathcal M_d(\mathbb C)$, $S\in\mathcal H(d)$ is a self-adjoint matrix and $\mathcal O_\mu$ is the set of $G\in \mathcal H(d)$ such that the eigenvalue list $\lambda(G)=\mu\in (\mathbb R^d)^\downarrow$.In this talk we describe the structure of local minimizers of $\Phi$, when we endow $\mathcal O_\mu$ with the metric induced by the spectral norm.The previous result plays a key role in the following context. Let $S\in\mathcal M_d(\mathbb C)^+$ be a positive semi-definite matrix, let$a=(a_i)_{i=1}^k\in (\mathbb R_{>0}^k)^\downarrow$ and consider the space $$\mathbb T_d(a)=\{\mathcal G=\{g_i\}_{i=1}^k\in (\mathbb C^d)^k:\ \|g_i\|^2=a_i\, , \ i=1,\ldots,k\}\,$$which is a product of spheres in $\mathbb C^d$, endowed with the product metric. We then consider the function$\Theta_{N,S,a}=\Theta:\mathbb T_d(a)\rightarrow \mathbb R_{\geq 0}$ given by $\Theta(\mathcal G)=N(S-S_\mathcal G)$, where $N$ is a uin and $S_\mathcal G=\sum_{i=1}^k g_i g_i^*M_d(\mathbb C)^+$ is the so-called frame operator of $\mathcal G$, for $\mathcal G=\{g_i\}_{i=1}^k\in \mathbb T_d(a)$.The function $\Theta$ is called a generalized frame operator distance.We will describe the structure of local minimizers of $\Theta$, show that these are global minimizers and compute the minimum value of $\Theta$.