Determine whether the matrix is positive/negative semidefinite/definite or indefinite:
$$\left(\begin{array}{cccc} 2&2&0&0\\ 2&2&0&0\\ 0&0&3&1\\ 0&0&1&3 \end{array}\right)$$ I already know that the answer is positive semidefinite because of the following statement:
Let $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{m\times m}$ be two symmetric matrices. Then the following are equivalent:
- $A$ and $B$ are positive semidefinite
- $C=\left(\begin{array}{cc}A&0_{n\times m}\\0_{m\times n}&B\end{array}\right)$ is positive semidefinite.
However, I want to find whether it is positive semidefinite or not in a different way. I know that a matrix is positive semidefinite if its eigenvalues are non-negative, but determining the eigenvalues of a $4\times 4$ matrix is quite tedious. Is there a different approach to this problem? Thanks.
