A Log-Det Inequality for Random Matrices
| Autor: | Rudolf Mathar, Lorens A. Imhof, Norbert Gaffke, Meik Dorpinghaus |
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| Rok vydání: | 2015 |
| Předmět: | |
| Zdroj: | SIAM Journal on Matrix Analysis and Applications. 36:1164-1179 |
| ISSN: | 1095-7162 0895-4798 |
| DOI: | 10.1137/140954647 |
| Popis: | We prove a new inequality for the expectation $E\left[\log\det\left(\mathbf{W}\mathbf{Q}+\mathbf{I}\right)\right]$, where $\mathbf{Q}$ is a nonnegative definite matrix and $\mathbf{W}$ is a diagonal random matrix with identically distributed nonnegative diagonal entries. A sharp lower bound is obtained by substituting $\mathbf{Q}$ by the diagonal matrix of its eigenvalues $\mathbf{\Gamma}$. Conversely, if this inequality holds for all $\mathbf{Q}$ and $\mathbf{\Gamma}$, then the diagonal entries of $\mathbf{W}$ are necessarily identically distributed. From this general result, we derive related deterministic inequalities of Muirhead- and Rado-type. We also present some applications in information theory: We derive bounds on the capacity of parallel Gaussian fading channels with colored additive noise and bounds on the achievable rate of noncoherent Gaussian fading channels. |
| Databáze: | OpenAIRE |
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