Computing eigenvalues and eigenvectors of a $2\times2$ matrix is easy by solving the characteristic equation. However, things get complicated if the matrix is larger.
Let's assume I have this matrix with computed eigenvalues and eigenvectors:
$$\begin{pmatrix}12&4\\3&7\end{pmatrix}$$
Then, I have this $4\times4$ matrix that contains two duplicates of this matrix in it:
$$\begin{pmatrix}12&4&0&0\\3&7&0&0\\0&0&12&4\\0&0&3&7\end{pmatrix}$$
To find the eigenvalues, I would have to solve an equation of $4$th degree and have to calculate a huge determinant. But I think there should be an easier way to calculate it.
I have $2$ questions here:
Is there a trick that I can use here to calculate them, knowing the eigenvalues of above $2\times2$ matrix already?
How would swapping the rows or columns of my $4\times4$ matrix change the eigenvalues?
Please feel free to answer any of the two. I am hoping that an easier solution exists to this.
