Eigenvalues of a $2 \times 2$ block matrix

Question

What are the eigenvalues of the following block matrix?

$$\begin{bmatrix} A & A \\ A & O \end{bmatrix}$$

Here, $A$ is any square matrix of order $n$ whose eigenvalues are $\lambda_1, \lambda_2, \dots, \lambda_n$ and $O$ is zero matrix of order $n$.

Answer 1

This is the Kronecker product $\pmatrix{1&1\\ 1&0}\otimes A$. Hence its eigenvalues are $\lambda_i\mu_j$ with $i=1,2,\ldots,n$ and $j=1,2$, where $\mu_1,\mu_2$ are the eigenvalues of $\pmatrix{1&1\\ 1&0}$.