Fractional Hadamard powers of symmetric positive-definite matrices

Abstract

Let A = (aij) andB = (bij) be matrices of the same size. Then their Hadamard product (also called Schur product) A B is dened by entrywise multiplication: A B = (aij bij) . The Hadamard unit matrix is the matrix U all of whose entries are 1 (the size of U being understood). A matrix A is Hadamard invertible if all its entries are non-zero, and A (= (a ij is then called the Hadamard inverse of A. IfB is Hadamard invertible, then the Hadamard quotient A =B of A and B is (aijb ij The k-fold Hadamard product A k of A with itself (k 0) is called the k-th Hadamard power of A; thus (aij) k = (a k ij ). In particular, A 0 = U (conventionally we set 0 0 = 1). If A is Hadamard invertible, then A k can be dened for negative integers as well, in an obvious manner. For more information on the Hadamard product, see [7, Chapter 5] and [5].