Determinant Inequalities via Information Theory

Simple inequalities from information theory prove Hadamard’s inequality and some of its generalizations. It is also proven that the determinant of a positive definite matrix is log-concave and that the ratio of the determinant of the matrix to the determinant of its principal minor $ | K_n | / | K_n - 1 |$ is concave, establishing the concavity of minimum mean squared error in linear prediction. For Toeplitz matrices, the normalized determinant $| K_n |^{1/n} $ is shown to decrease with n.