Some inequalities for sums of matrices (Q2758096)

From a generalized matrix version of the Cauchy-Schwarz and Fouch-Kantorovich inequalities NEWLINE\[NEWLINE (X^{\ast}AX)^{-1}\leq X^{\ast}A^{-1}X \leq \frac{(\lambda_1+ \lambda_2)^2}{4\lambda_1\lambda_2}(X^{\ast}AX)^{-1} NEWLINE\]NEWLINE where \(A\) is an \(n\times n\) Hermitian positive definite matrix with eigenvalues \(\lambda_1 \geq \lambda_2\geq \cdots\lambda_n>0\), and the \(n\times p\) complex matrix \(X\) and its conjugate transpose \(X^{\ast}\) satisfy \(X^{\ast}X=I_p\), some inequalities are given involving the Moore-Penrose inverse. In the above inequality the Löwner partial ordering \(M\geq N\) means \(M-N\) is Hermitian nonnegative definite. NEWLINENEWLINENEWLINERelated to the first inequality a result is given for sums of \(k\) Hermitian nonnegative definite matrices of order \(n\) and \(k\) complex matrices of size \(n\times p\). In addition, the equality is characterized. Finally, another inequality corresponding to the right hand side of the above inequality is obtained for the sum of \(k\) Moore-Penrose matrices of nonnull Hermitian nonnegative definite matrices.