Prove: if $B$ is semidefinite, then $B_{11}$ is semidefinite as well

Question

If $$B = \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{bmatrix}$$ is a symmetric positive semidefinite matrix of order $p$, prove that its leading principal submatrix $B_{11}$ of order $q$ is also positive semidefinite.

Answer 1

Note that $B$ is positive (negative) semidefinite iff $x^TBx \geq 0$ ($x^TBx \leq 0$) for every vector $x$.

Now, let $x$ be the "block vector" $x = \pmatrix{y\\0}$. We note that $$ x^TBx = \pmatrix{y^T&0} \pmatrix{B_{11}&B_{12}\\B_{21}&B_{22}} \pmatrix{y\\0} = y^T B_{11} y $$

Answer 2

Note that $B_{11}$ is a symmetric submatrix of a symmetric matrix $B$ and apply Cauchy interlacing theorem.