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Hi I have a matrix of the following form arising by discretization of a system of PDEs. I am working to get the invertibility of the Matrix. Can some one help me or at least give me some reference on these type of problem?
$A=\begin{bmatrix} A_{1,1} & A_{1,2} & \cdots & A_{1,n} \\ A_{2,1} & A_{2,2} & \cdots & A_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ A_{n,1} & A_{n,2} & \cdots & A_{n,n} \end{bmatrix}$
where each $A_{i,j}$ is a symmetric and positive definite matrix. Each have diagonal term strictly positive and non-diagonal terms $\leq 0$ and diagonally dominant(not strictly). Moreover $A_{i,j}=A_{j,i},\ \forall i,j$. Can someone give any idea to prove whether matrix $A$ is positive definite or at least invertible? May be by using the proof of $M$-matrix in the single matrix case or some other method??? $A$ will not be $M$-Matrix because diagonal element of some $A_{i,j}>0,i\neq j$.

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Your matrix can be singular if all blocks are identical. More conditions are needed to ensure invertibility.

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