An inequality for positive definite matrices with applications to combinatorial matrices
Let \(A\in M_n(\mathbb C)\) be a positive definite matrix. Let \(d,f,f'\) stand for the averages of the diagonal, off-diagonal, and modified off-diagonal entries \(a_{ij}/\sqrt{a_{ii}a_{jj}}\) of \(A\), respectively. Then \(|\det(A)|\leq (d-f)^{n-1}[d+(n-1)f]\), equivalently, \(|\det(A)|\leq (1-f')^{n-1}[1+(n-1)f']\prod_{i=1}^n a_{ii}\), a strengthening of Hadamard's inequality for positive definite matrices. These results are applied to the case the positive definite matrix is obtained as \(AA^T\) from some rectangular matrix \(A\), in particular for the case \(A\) being from one of these classes: entrywise nonnegative, stochastic, \((0,1)\), or \((\pm 1)\). In the latter two cases inequalities of e.g. \textit{H. J. Ryser} [Can. J. Math. 8, 245-249 (1956; Zbl 0071.35903)], \textit{H. Ehlich} [Math. Z. 83, 123-132 (1964; Zbl 0115.24704)], and \textit{C.-S. Cheng} [Ann. Stat. 8, 436-446 (1980; Zbl 0425.62055)] are recovered.
scientific article; zbMATH DE number 1089495
scientific article