Positive definiteness of block matrices I have a $2 \times 2$ block matrix of the form $$M = \left[ {\begin{array}{*{20}{c}}
{\delta I}&A\\ {{A^T}}&kA \end{array}} \right]$$ where the matrix $A$ is positive definite and not symmetric, $I$ is the identity matrix, and $k > 0$ and $\delta > 0$.

*

*Can we choose $\delta > 0$ such that the matrix $M$ be positive definite?


*Is there any general formula for positive definiteness of block matrices? It seems that the Schur complement is only for symmetric matrices.
I really appreciate if anyone can help me regarding this problem.

I find a solution but not sure if it's correct.
Consider 
$S=[x_1 \hspace{.5cm} x_2]M \left[ \begin{array}{l}
{x_1}\\
{x_2}
\end{array} \right]=\delta x_1^T{x_1} + 2x_1^TA{x_2} + kx_2^TA{x_2}$
-> $S \le \delta {\left\| {{x_1}} \right\|^2} - {2\sigma _{\max }}(A)\left\| {{x_1}} \right\|\left\| {{x_2}} \right\| + k{\lambda _{\min }}(A){\left\| {{x_2}} \right\|^2} $
Hence, if we have 
$\delta k{\lambda _{\min }}(A) - {4\lambda _{\max }}({A^T}A) \ge 0 \Rightarrow \delta  \ge \frac{{{4\lambda _{\max }}({A^T}A)}}{{k{\lambda _{\min }}(A)}}$
then $S\ge0$.
