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 | Kₙ | / | Kₙ - 1 | is concave, establishing the concavity of minimum mean squared error in linear prediction. For Toeplitz matrices, the normalized determinant | Kₙ |1/n is shown to decrease with n.