Relation between positive definite Hermitian matrices with their inverses

Question

Let $A$ and $B$ be two positive definite Hermitian matrices. Show that the Hermitian matrix $$C\ =\ A^{-1} + B^{-1} - 4(A + B)^{-1}$$ is also positive definite.

Thanks in advance.

Answer 1

The statement is not true. Consider $A=B=I$. Then $C=0$ is not positive definite.

Edit: Let $A^{1/2}$ be the positive definite square root of $A$ and $X=A^{-1/2}BA^{-1/2}$. Then $$C=A^{-1/2}[I+X^{-1}-4(I+X)^{-1}]A^{-1/2}.$$ Therefore $C$ is positive definite if and only if $I+X^{-1}-4(I+X)^{-1}$ is positive definite. Since $X$ is positive definite, it is unitarily diagonalizable. Therefore WLOG we may assume that $X$ is a positive diagonal matrix. So it boils down to figuring out when will $1+x^{-1}-4(1+x)^{-1}>0$ for $x>0$. As this inequality is true for all $x\neq1$ and equality holds when $x=1$, we conclude that $C$ is always positive semi-definite and it is positive definite if and only if $1$ is not an eigenvalue of $X=A^{-1/2}BA^{-1/2}$. (By the way, the eigenvalues of $A^{-1/2}BA^{-1/2}$ are identical to the eigenvalues of $A^{-1}B$ or $BA^{-1}$.)