POTW Positive Definite Block Matrices

[Euge]

Suppose $A$ and $B$ are positive definite complex $n \times n$ matrices. Let $M$ be an arbitrary complex $n \times n$ matrix. Show that the block matrix $\begin{pmatrix} A & M\\ M^* & B\end{pmatrix}$ is positive definite if and only if $M = A^{1/2}CB^{1/2}$ for some matrix $C$ of operator norm $\|C\| < 1$.

 

[Euge]

Spoiler

The matrix $\begin{pmatrix}A & M\\M^* & B\end{pmatrix}$ is positive definite if and only if
$$\begin{pmatrix}A^{-1/2} & 0\\0 & B^{-1/2}\end{pmatrix} \begin{pmatrix}A & M\\M^* & B\end{pmatrix} \begin{pmatrix} A^{-1/2} & 0\\0 & B^{-1/2}\end{pmatrix} = \begin{pmatrix} I & A^{-1/2}M B^{-1/2}\\ B^{-1/2}M^* A^{-1/2} & I\end{pmatrix}$$ is positive definite. Let $C = A^{-1/2} M B^{-1/2}$, and write $C$ in SVD $C = U\Sigma V^*$ where $\Sigma = \operatorname{diag}(\sigma_1,\ldots, \sigma_n)$ is the diagonal matrix of singular values of $C$. Since $$\begin{pmatrix}I & C\\C^* & I\end{pmatrix} = \begin{pmatrix}U & 0\\0 & V\end{pmatrix} \begin{pmatrix} I & \Sigma\\ \Sigma & I\end{pmatrix} \begin{pmatrix}U^* & 0 \\0 & V^*\end{pmatrix}$$ then $\begin{pmatrix} I & C\\C^* & I\end{pmatrix}$ is positive definite if and only if $\begin{pmatrix}I & \Sigma\\\Sigma & I\end{pmatrix}$ is positive definite. The latter matrix is unitarily similar to block sum $$\bigoplus_{i = 1}^n \begin{pmatrix}1 & \sigma_i\\\sigma_i & 1\end{pmatrix}$$ It follows that $\begin{pmatrix}I & C\\C^* & I\end{pmatrix}$ is positive definite if and only if $\sigma_i^2 < 1$ for all $i$, i.e., $\|C\| < 1$. Finally, observe that the equation for $C$ is equivalent to $M = A^{1/2} C B^{1/2}$.