$ \begin{bmatrix} D_1 & C \\ C^T & D_2 \\ \end{bmatrix} $ always positive semidefinite?

Question

Is block matrix of the form $$ \begin{bmatrix} D_1 & C \\ C^T & D_2 \\ \end{bmatrix} $$, where $C$ is a nonnegative matrix (entrywise nonnegative) and $D_1$ and $D_2$ are diagonal matrix also nonnegative, always positive semidefinite?

If we choose $D_1$ and $D_2$ such that the matrix is diagonally dominant, then we can prove it!!

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Let $x$ be any vector, partitioned suitably as $x = (x_1\ x_2)^T$. Then $$ \begin{bmatrix} x_1^T & x_2^T \end{bmatrix} \begin{bmatrix} D_1 & C \\ C^T & D_2 \\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = x_1^TD_1x_1 + x_1^T Cx_2+x_2^TC^Tx_1+x_2^TD_2x_2 $$

Then how should I proceed?

Answer 1

Consider the matrix $\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}$, the determinant is $-1$. Hence it is not positive semidefinite.