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 $\epsilon \le 1/n$, then a lower bound of Ostrowski (1938) is $\det (A)\ge 1-n\epsilon$. 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 $\epsilon <1/n$) $\det (A)\le (1-n\epsilon {)}^{-1}$. The second upper bound generalises Hadamard's inequality, which is the case $\epsilon =1$. A necessary and sufficient condition for our upper bounds to be best possible for matrices of order $n$ and all positive $\epsilon$ is the existence of a skew-Hadamard matrix of order $n$.

18 pages, 39 references. Added some references in v7

References in corpus (1)

• Lower bounds on maximal determinants of binary matrices via the probabilistic method

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