# How to find eigenvalues of following block matrices?

Is there a procedure to find the eigenvalues of A? ‎

$$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^t & 0 & I &&&&&&&\\&&I & 0 & P &&&&&&&\\&&& P^t & 0 & \ddots &&&&&\\&&&& \ddots & \ddots & \ddots &&&&\\&&&&& \ddots & 0 & I &&&\\&&&&&& I & 0 & P &&\\&&&&&&& P^t & 0 & I &0_{k\times y-k}\\&&&&&&&& I &&\\0&&&&&&&& 0_{y-k\times k} && Y_{y\times y}\end{bmatrix}$$ (All rows and columns except the last are $$k \times k$$ blocks.)

where $$A$$ is adjacency matrix of cubic graphs and $$X, P$$ are circulant matrices of order k, Y is a matrix of order y, $$k \not=y$$ and $$I$$ is a identity matrix and $$0$$ is a zero matrix.

For example: How to find eigenvalues of following block matrices?

• – Rodrigo de Azevedo Jun 20 '18 at 7:56
• You mean efficiently or analytically? What's the problem with the regular eig()? – percusse Jun 25 '18 at 16:15
• I want eigenvalues or characteristic polynomial associated to a cubic (3-regular) graphs with adjacency matrix A. Since $y\not= k$, then A is not Jacobian three-diagonal matrix and I cannot find characteristic polynomial of A. – Maryam Hak Jun 25 '18 at 21:12