eigenvalue of overlapping block matrix

eegyan

I have got a problem in my research. For the following matrix,
a a a a a a b b b b
a a a a a a b b b b
a a a a a a b b b b
a a a a a a b b b b
a a a a a a a a a a
a a a a a a a a a a
b b b b a a a a a a
b b b b a a a a a a
b b b b a a a a a a
b b b b a a a a a a,
does anyone know how to obtain its eigenvalues analytically when the matrix size goes large? Is there any "name" for matrices just like this in mathmatics? I wanna search for some inspiration in mathmatics matrix community, but I even do not know what it is called in mathmatics. Thanks in advance.

Office_Shredder

Well the matrix is rank 3, so all but three of the eigenvalues are zero. We can use the block structure to find the eigenvectors, and hence the eigenvalues.

If there are x, y and z of each type of row, a vector with x ones then all zeros is an eigenvector, as is a vector with x zeros, y ones then z zeros, and so is a vector with x+y zeros followed by z ones