On fractional Hadamard powers of positive definite matrices

Critical exponents: old and new

“…That n − 2 is also the lower bound for the critical exponent follows from the following example, provided in [5]. Define A = (1 + ij), 1 ≤ i, j ≤ n. Analysis in the next section will verify that A is DN, and the vectors v k = (1 k , 2 k , · · · , n k ) T for k = 0, 1, 2, · · · can be determined to be linearly independent because the determinant of the Vandermonde matrix produced from them is positive.…”

“…A detailed discussion of the original proof can be found in [5], although the topic is also covered in [6].…”

“…The proof by induction of this statement appears in [5] and [6] and involves a modified version of the formula for the remainder of a Taylor Series. Specifically, let…”

“…Because γγ T and A − γγ T are both positive semi-definite matrices (verified in [5]), the integral is the limit of Riemann sums of PSD matrices raised to the (t − 1)-th power. Hadamard multiplication by A − γγ T selects the upper-left (n − 1)-by-(n − 1) submatrix of every matrix in this sum which, by the induction hypothesis, is DN for all t ≥ n − 2.…”

The Schur Product Theorem in the Block Case

“…The special case n = r = \ leads to consideration of Hadamard products and functions, about which much is known [1,2,6]. John de Pillis [5] showed that if H is positive semi-definite and /(■#,-.)…”