Let $A$ be a square, symmetric and positive definite matrix, now, consider the following block matrix

$$M=\begin{bmatrix}2A&A&A&\cdots&A\\A&2A&A&\cdots&A\\A&A&2A&\cdots &A\\ \vdots&\vdots&\vdots&\ddots&\vdots\\A&A&A&\cdots&2A\end{bmatrix}$$

Here, $M$ is a square block matrix and its entries are $2A$ in the diagonal and $A$ in any other position.

How can I prove that the block matrix $M$ is positive definite?

I already prove this for the cases when $M$ is a $2\times 2$ and $3\times 3$ block matrix using the Schur complement theorem.


1 Answer 1


Hint: $$\begin{bmatrix}x_1^\top&x_2^\top&\dots&x_n^\top\end{bmatrix}\begin{bmatrix}2A&A&A&\cdots&A\\A&2A&A&\cdots&A\\A&A&2A&\cdots &A\\ \vdots&\vdots&\vdots&\ddots&\vdots\\A&A&A&\cdots&2A\end{bmatrix}\begin{bmatrix}x_1\\x_2\\ \vdots\\ x_n \end{bmatrix}\\ = \sum_\limits{i=1}^n x_i^\top Ax_i +\left(\sum_\limits{i=1}^nx_i\right)^\top A\left(\sum_\limits{i=1}^nx_i\right)$$


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