Consider the block matrix given by $$M = \begin{bmatrix} A_{11}&A_{12}&0\\ A_{12}&A_{22}&A_{23}\\ 0&A_{23}&A_{33}\end{bmatrix}$$

What conditions should I impose on each matrix $A_{ij}$ to ensure that M is positive definite? Each matrix $A_{ij}$ is symmetric. I would like not to use a Schur complement-like condition to avoid deal with inverses. Also, could someone provide a good reference on this subject?

Thank you

  • 1
    $\begingroup$ If $M$ is PD, its principal submatrices have to be PD too. So, the first question that you should ask yourself is: how to decide whether $\pmatrix{A_{11}&A_{12}\\ A_{12}&A_{22}}$ is PD without using Schur complements? $\endgroup$ – user1551 Jul 12 '16 at 11:17
  • $\begingroup$ Actually, my question was the opposite. Formulating in terms of the principal submatrices would be like this. What are the conditions over the principal submatrices for which M is PD? Showing that these submatrices are PD is enough? $\endgroup$ – Cybernetician Jul 12 '16 at 11:32
  • $\begingroup$ $\left(\begin{array}{c} A & X \\ X^* & B \end{array}\right)$ is positive iff $X=A^{1/2}K B^{1/2}$ where $K$ is a contraction $\endgroup$ – Hamza Jul 12 '16 at 19:57
  • $\begingroup$ How do you define a contraction? $\endgroup$ – Cybernetician Jul 13 '16 at 0:39

Based on the post by user1551, I came up with the following proof.

If each matrix $A_{ij}$ is such that the block submatrices

\begin{equation*} M_1=\begin{bmatrix} A_{11}&A_{12}\\ A_{12}&A_{22}/2 \end{bmatrix}, M_2=\begin{bmatrix} A_{22}/2&A_{23}\\ A_{23}&A_{33} \end{bmatrix} \end{equation*} is positive definite, then the matrix $M$ is positive definite.

Proof. By assumption $M_1$ and $M_2$ are positive definite. Consequently, for vectors $(x,y)$ and $(y,z)$ the following inequalities hold \begin{align*} x^\top A_{11}x + x^\top A_{12}y + y^\top A_{12} x + y^\top \dfrac{A_{22}}{2} y&>0\\ y^\top \dfrac{A_{22}}{2}y + y^\top A_{23}z + z^\top A_{23} y + z^\top A_{33} z&>0 \end{align*} Summing both equations yields \begin{equation} x^\top A_{11}x + x^\top A_{12}y + y^\top A_{12} x + y^\top A_{22}y+y^\top A_{23}z + z^\top A_{23} y + z^\top A_{33} z>0. \end{equation} This inequality implies that the matrix $M$ is positive definite.


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