# Question on positive definite block matrix.

Suppose that we have symmetric positive definite matrices $A,B,D \in R^{n \times n}$. Further, we have $$\begin{bmatrix}A&C\\C^T &B \end{bmatrix} > 0$$ and also $$\begin{bmatrix}A B^{-1} A &C\\C^T & B A^{-1} B \end{bmatrix} > 0$$ holds.

Does anyone knows whether $$\begin{bmatrix}(A+D) (B+D)^{-1} (A+D) & C+D\\C^T +D & (B+D) (A+D)^{-1} (B+D) \end{bmatrix} > 0$$ holds? Thank you!

• Have you done any computer experiment to see that if this conjecture is convincing or not? Jan 13, 2015 at 20:27
• I did some tests and it seems to work, but I do not know how to show this.
– Ben
Jan 14, 2015 at 14:15

The conjecture is wrong. It should be easy to generate a random counterexample using a computer. E.g. I obtained the followings: $$\begin{bmatrix}A&C\\ C^T&B\end{bmatrix}= \left[\begin{array}{cc|cc} 7&1&0&0\\ 1&1&0&2\\ \hline 0&0&3&1\\ 0&2&1&7 \end{array}\right], \ D=\begin{bmatrix}5&-3\\ -3&3\end{bmatrix}.$$ you may verify that $A,B,D$ as well as the first two block matrices in your question are positive definite, but the last block matrix in your question has a negative eigenvalue.