On the Range of the Hadamard Product of a Positive Definite Matrix and Its Inverse

Abstract

Suppose A is an $n \times n$ real positive definite matrix. This paper characterizes for $n = 3$ the range of the operator $A \circ A^{ - 1} $, where $ \circ $ denotes the Hadamard product. It is shown that the necessary conditions exhibited by the first author in 1964 [M. Fiedler, Czechoslovak Math. J., 14 (1964), pp. 39–51] are also sufficient in this case.