How do I find the eigenvalues of the matrix below,

0 A 0 0 0 ...
A' 0 A 0 0 ...
0 A' 0 A 0 ...
0 0 A' 0 A ...
...

where A is a 2x2 matrix

1 k
k 1

and A' is also a 2x2 matrix

1 -k
-k 1

Do the eigenvalues of the whole matrix depend on the eigenvalues of the block matrices A and A'?