<s>[INST] Questions regarding minors of a positive definite matrix In my lectures on Matrix computations, there's a section titled Gaussian elimination and Cholesky decomposition. It is a follows:
suppose $A=A^T$ is positive definite, $a_{11>0}$ and $A_{22}$ is positive definite as well, then we have:
$
A=\begin{bmatrix}
a_{11} & A_{21}^T\\
A_{21} & A_{22}\\
\end{bmatrix}
$
$
E_{1}=\begin{bmatrix}
1 & 0\\
-A_{21}.a_{11}^{-1} & I_{n-1}\\
\end{bmatrix}
$
$\implies$
$
E_1A=\begin{bmatrix}
a_{11}&A_{21}^T\\
0&A_{22}-A_{21}.a_{11}^{-1}.A_{21}^T
\end{bmatrix}
=A^{(2)}
$
I don't seem to understand why $A_{22}$ has to be positive definite. And from there, why is $A_{22}-A_{21}.a_{11}^{-1}.A_{21}^T$ positive definite, I don't get it. 
 [/INST] For the other expression see that for $u = (x, Y^T)^T \neq 0$ where $Y\in \mathbb R^{n-1}$ and $x\in\mathbb R$ we have
$$\begin{align*}u^T A u & = (x\quad Y^T) \left(\begin{matrix} xa_{11}+ A_{21}^T Y \\ xA_{21} + A_{22} Y\end{matrix}\right) \\
& = x^2a_{11} + x A_{21}^T Y + Y^T A_{21} x + Y^T A_{22} Y \\
& =x^2a_{11} + 2xA_{21}^T Y + Y^T A_{22} Y > 0 \end{align*}$$
If you chose $x = 0$ and $Y \neq 0$ you have $A_{22} \succ 0$.  

For $x = -A_{21}^TY \cdot a_{11}^{-1}$ and $Y \neq 0$ we get
$$\begin{align*}
\underbrace{Y^T A_{21} a_{11}^{-2} A_{21}^T Y}_{=x^2} \cdot a_{11} \underbrace{-2a_{11}^{-1} (A_{21}^TY)^2}_{=2 x A_{21}^T Y} + Y^T A Y & > 0 \\
-a_{11}^{-1} (A_{21}^TY)^2 + Y^T A_{22} Y & > 0 \\
Y^T (A_{22} - a_{11}^{-1} A_{21} A_{21}^T)Y & > 0\end{align*}$$
And you're done.
</s>