Majorization relations for Hadamard products (Q1894481)

\textit{C. R. Johnson} and \textit{R. B. Bapat} [Linear Algebra Appl. 104, 246- 247 (1988)] have conjectured that: if \(A\) and \(B\) are \(n \times n\) positive definite matrices with Hadamard product \(A \circ B\) then, for each \(k \leq n\), the product of the \(k\) smallest of the eigenvalues of \(A \circ B\) is at least as great as the product of the \(k\) smallest of the eigenvalues of \(AB\). This paper provides a proof of this conjecture and shows that the corresponding inequalities for the eigenvalues of \(A \circ B\) and \(AB^T\) and for the eigenvalues of \(A \circ B\) and the ``geometric mean'' \(A \# B : = A^{1/2} (A^{-1/2} BA^{-1/2})^{1/2} A^{1/2}\) are also true. \{Another proof of the Bapat-Johnson conjecture has been given by \textit{G. Visick}, ``A weak majorization involving the matrices \(A \circ B\) and \(AB\)'', Linear Algebra, Appl. 223-224, 731-744 (1995)\}.