Eigenvalues of a block diagonal symmetric matrix

Question

If we hava a block diagonal matrix $M$, \begin{equation} M= \begin{bmatrix} d_{1} & & \\ & \ddots & \\ & & d_{1} \end{bmatrix} \end{equation} where $d_{1}$ is ($4 \times 4$) symmetric matrix, \begin{equation} d_{1}= \begin{bmatrix} a & b & c & b \\ b & a & b & c \\ c & b & a & b \\ b & c & b & a \end{bmatrix} \end{equation}

then how we will find the eigenvalues of $M$ ?

Answer 1

The eigenvalues of $M$ will be the eigenvalues of $d_1$. It turns out that the eigenvalues of $d_1$ are $a-c$ (with multiplicity $2$) and $a+c\pm2b$.