An inequality for positive definite matrices with applications to combinatorial matrices (original) (raw)

Linear Algebra and its Applications, 1990

We estimate the di%xence of determinants of two square matrices in terms of the norms of their difference.

Establishing determinantal inequalities for positive-definite matrices

Discrete Applied Mathematics, 1995

Let A be an n x n matrix, and S be a subset of N = { 1,2, . . . . n}. A[S'] denotes the principal submatrix of A which lies in the rows and columns indexed by S. If M. = {al, . . ..a.} and B = I&? . ..TB.} are two collections of subsets of N, the inequality t( < /? expresses that flp= 1 det A[aJ 6 ns= 1 det A[ pi], for all n x n positive-definite matrices A.

Probabilistic lower bounds on maximal determinants of binary matrices

Let D(n) be the maximal determinant for n × n {±1}-matrices, and R(n) = D(n)/n n/2 be the ratio of D(n) to the Hadamard upper bound. Using the probabilistic method, we prove new lower bounds on D(n) and R(n) in terms of d = nh, where h is the order of a Hadamard matrix and h is maximal subject to h ≤ n. For example, By a recent result of Livinskyi, d 2 /h 1/2 → 0 as n → ∞, so the second bound is close to (πe/2) -d/2 for large n. Previous lower bounds tended to zero as n → ∞ with d fixed, except in the cases d ∈ {0, 1}. For d ≥ 2, our bounds are better for all sufficiently large n. If the Hadamard conjecture is true, then d ≤ 3, so the first bound above shows that R(n) is bounded below by a positive constant (πe/2) -3/2 > 0.1133.

Inequalities for Spreads of Matrix Sums and Products

Applied Mathematics E-Notes, 2004

Let A and B be complex matrices of same dimension. Given their eigen-values and singular values, we survey and further develop simple inequalities for eigenvalues and singular values of A + B, AB, and A ◦ B. Here ◦ denotes the Hadamard product. As corollaries, we ...