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Let's say we have two positive semi-definite matrices $A$ and $B$ and a negative real $\lambda$. What would be the conditions for the matrix $M$, defined as follows, to be positive semi-definite too ? $$M = \begin{pmatrix} A & \lambda I\\ \lambda I & B \end{pmatrix},$$ where $I$ is the identity matrix.

I know that $M$ is positive semi-definite if and only if its Schur complement $S=A- \lambda^2 B$ is positive semi-definite, as it is stated here. But it doesn't help me much... Furthermore, I would like to find conditions which work also for block matrices with more than 2 blocks per row and columns like this one: $$\begin{pmatrix} A & \lambda_{1,2} I & \lambda_{1,3} I \\ \lambda_{1,2} I & B & \lambda_{2,3} I \\ \lambda_{1,3} I & \lambda_{2,3} I & C \end{pmatrix},$$

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  • $\begingroup$ If you didn't like the answer for the four block case, you won't like this answer for the nine-block case. But you could go through the steps of the Cholesky Factorization algorithm, and obtain a pair of conditions like the one you had above (but one of the conditions would be rather more complicated to write down). $\endgroup$ – Stephen Montgomery-Smith Mar 5 '14 at 13:36

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