Q:

Prove that a blockmatrix is positive definite

Let $A, B \in \mathbb{R}^{n \times n}$ be symmetric and positive definite matrices. Define: $$C = \begin{pmatrix} A_{11}B & A_{12}B & \cdots & A_{1n}B\\
\vdots & \vdots & \ddots & \vdots \\ A_{n1}B & A_{n2}B & \cdots & A_{nn}B \end{pmatrix} \in \mathbb{R}^{n^2 \times n^2}$$
Show that $C$ is symmetric and positive definite. 
The proof that $C$ is symmetric is simple since it is a direct consequence of $A^T = A$ and $B^T = B$. The problem is to prove that $C$ is positive definite.
Let $x \in \mathbb{R}^{n^2}\setminus \{0\}$ with $x = (x_1,\dots,x_n)^T$ and $x_i \in \mathbb{R}^n$. I'd like to reduce the problem of $x^TCx > 0$ so that I can use the assumptions. Computing the regular formula for $x^TCx$ ended in a chaos for me, is there any simpler way to prove this?

A:

This is called the Kronecker product and is denoted by $A\otimes B$.
One can show that the eigenvalues of $C$ are given by:
$$\{\lambda_C\}=\{\lambda_A\}\{\lambda_B\}$$
That is, eigenvaues of $C$ are all products of eigenvalues of $A$ and $B$.
So for $\{\lambda_A\},\{\lambda_B\}>0$, we get $\{\lambda_C\}>0$.
To prove:
$$\begin{align} (A\otimes B)(x\otimes y)&=Ax\otimes By\\
&=\lambda_Ax\otimes\lambda_By\\
&=\lambda_A\lambda_B(x\otimes y)
\end{align}$$
It is also not difficult to show that 
$$(A\otimes B)^T=A^T \otimes B^T$$
which can be used to conclude that the Kronecker product is a symmetric matrix when $A$ and $B$ are symmetric.

