Let $A$ be a positive definite matrix partitioned in the following fashion.
$$A=\begin{bmatrix} A_{11} & A_{12} \\ A_{12}^T & A_{22}\end{bmatrix}.$$
In particular, $A_{11}$ is $m_1\times m_1$ and $A_{22}$ is $m_2\times m_2$. We need to show that both $A_{11}$ and $A_{22}$ are positive definite.
My Attempt. Let $x=[x_1^T\quad x_2^T]^T$ be an $(m_1+m_2)\times 1$ vector, where $x_1$ is $m_1\times 1$ and $x_2$ is $m_2\times 1$. Then
$$x^TAx=x_1^TA_{11}x_1+x_2^TA_{22}x_2+2x_1^TA_{12}x_2.$$
We know that $x^TAx>0$ whenever $x\neq 0$. However, we cannot make sure both $x_1^TA_{11}x_1>0$ and $x_2^TA_{22}x_2>0$, because I think it is possible that
$$x_1^TA_{11}x_1=0,\quad x_2^TA_{22}x_2=0,\quad x_1^TA_{12}x_2>0.$$
Perhaps I missed something here. Hope anyone have good ideas.
Best Answer
Hint: We know that $x^TAx > 0$ for every $x$ with $x_1 \neq 0$ and $x_2 = 0$. What does this tell you about $A_{11}$?