Hadamard inverses, square roots and products of almost semidefinite matrices

Abstract

Let A = (a ij) be an n × n symmetric matrix with all positive entries and just one positive eigenvalue. Bapat proved then that the Hadamard inverse of A, given by A •(−1) = (1 a ij) is positive semidefinite. We show that if moreover A is invertible then A •(−1) is positive definite. We use this result to obtain a simple proof that with the same hypotheses on A, except that all the diagonal entries of A are zero, the Hadamard square root of A, given by A • 1 2 = (a 1 2 ij), has just one positive eigenvalue and is invertible. Finally, we show that if A is any positive semidefinite matrix and B is almost positive definite and invertible then A • B

On the Bounds for the Norms of R-Circulant Matrices With the Jacobsthal and Jacobsthal Lucas Numbers

“…For any A, B ∈ M m,n (C) , the Hadamard product of A, B is entrywise product and defined by (see [5,6])…”

Conditional Quadratic Semidefinite Programming: Examples and Methods

“…17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Proposition 4.5 Let h ∈ IR m be given and Q T (A * (h))Q have the representation of (35). Let A ∈ K n + (V) be decomposed as in (37) and the resulting Z 1 have the spectral decomposition (38). Constraint nondegeneracy holds at A if and only if the following implication holds…”

“…For example, in the survey paper [24], Ikramov and Savel'eva used it for V being not only a subspace but also a closed convex cone. Reams [38] used it for V = e ⊥ , the subspace orthogonal to the vector e of all ones. Earlier, Chabrillac and Crouzeix in their survey paper [7] call it the semidefiniteness of the restricted quadratic form to the space V. Micchelli [29] (see also Baxter [3]) used a slightly different name for V = e ⊥ .…”

Synchronization optimization of networked second order infinite dimensional systems

“…The matrix L •(−1) denotes the restricted Hadamard inverse [7] of L. The restricted Hadamard inverse of L is the matrix whose i jth nonzero entry is at the same location as the i jth nonzero entry of L and whose value is equal to 1/L i j .…”