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?

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.