A LOG-DET INEQUALITY FOR RANDOM MATRICES

作者：Doerpinghaus Meik; Gaffke Norbert; Imhof Lorens A; Mathar Rudolf

来源：SIAM Journal on Matrix Analysis and Applications, 2015, 36(3): 1164-1179.

DOI：10.1137/140954647

• expectation inequality
• rearrangement inequality
• Muirhead-type inequality
• Rado-type inequality
• L-superadditive function
• conditional entropy
• channel capacity

We prove a new inequality for the expectation E [log det (WQ + I)], where Q is a nonnegative definite matrix and W is a diagonal random matrix with identically distributed nonnegative diagonal entries. A sharp lower bound is obtained by substituting Q by the diagonal matrix of its eigenvalues Gamma. Conversely, if this inequality holds for all Q and Gamma, then the diagonal entries of 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.

• 出版日期2015