Bounds on determinants of perturbed diagonal matrices

Abstract

We give upper and lower bounds on the determinant of a perturbation of the identity matrix or, more generally, a perturbation of a nonsingular diagonal matrix. The matrices considered are, in general, diagonally dominant. The lower bounds are best possible, and in several cases they are stronger than well-known bounds due to Ostrowski and other authors. If $A = I-E$ is an $n \times n$ matrix and the elements of $E$ are bounded in absolute value by $\varepsilon \le 1/n$, then a lower bound of Ostrowski (1938) is $\det(A) \ge 1-n\varepsilon$. We show that if, in addition, the diagonal elements of $E$ are zero, then a best-possible lower bound is \[\det(A) \ge (1-(n-1)\varepsilon)\,(1+\varepsilon)^{n-1}.\] Corresponding upper bounds are respectively \[\det(A) \le (1 + 2\varepsilon + n\varepsilon^2)^{n/2}\] and \[\det(A) \le (1 + (n-1)\varepsilon^2)^{n/2}.\] The first upper bound is stronger than Ostrowski's bound (for $\varepsilon < 1/n$) $\det(A) \le (1 - n\varepsilon)^{-1}$. The second upper bound generalises Hadamard's inequality, which is the case $\varepsilon = 1$. A necessary and sufficient condition for our upper bounds to be best possible for matrices of order $n$ and all positive $\varepsilon$ is the existence of a skew-Hadamard matrix of order $n$.