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

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!!

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?

• Is $$\pmatrix{1&7\cr7&1\cr}$$ positive semidefinite? – Gerry Myerson Oct 15 '18 at 3:45
• Any thoughts on the answer/comment? – Gerry Myerson Oct 17 '18 at 0:58
• Are you still here, userly? – Gerry Myerson Oct 18 '18 at 1:19

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