Abstract
If A ∈ M n( C) is a positive definite Hermitian matrix, d the average of the diagonal entries of A, and f the average of the absolute values of the off-diagonal entries of A, then det A ⩽ ( d − f) n−1 [ d + ( n − 1) f]. As a corollary we obtain a strengthening of Hadamard's inequality for positive definite matrices. The results can be used to prove inequalities for the determinants of (± 1) matrices, (0, 1) matrices, positive matrices, stochastic matrices, and constant-column-sum matrices.
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