Finding the eigenvalues of a symmetric matrix

I want to find the eigenvalues of the matrix \begin{pmatrix} 5 & -1 & -1 & -1\\ -1 & 5 & -1 & -1\\ -1 & -1 & 5 & -1\\ -1 & -1 & -1 & 5\\ \end{pmatrix}

in a short way. The hint said that it is easy if I can write the matrix as the sum of two matrices.

Could someone explain to me the hint?

Thanks everyone for reading.

$$ \begin{pmatrix} 5& -1 & -1 &-1 \\ -1 & 5 & -1&-1 \\ -1 & -1 & 5 &-1 \\ -1 & -1 & -1 &5 \end{pmatrix} = \begin{pmatrix} -1& -1 & -1 &-1 \\ -1 & -1 & -1&-1 \\ -1 & -1 & -1 &-1 \\ -1 & -1 & -1 &-1 \end{pmatrix} + \begin{pmatrix} 6& 0 & 0 &0 \\ 0 & 6 & 0&0 \\ 0 & 0 & 6 &0 \\ 0 & 0 & 0 &6 \end{pmatrix} $$

It is easy to see that the first matrix in the sum decomposition is of rank 1, thus it has the eigenvalue 0 with multiplicity 3. The last eigenvalue can be computed in several way, but the easiest one is using the trace of the matrix, since the trace equals the sum of all the eigenvalues we obtain directly that the last eigenvalue is -4.

About the second matrix, it is a diagonal matrix with a constant diagonal thus the entire space is the eigenspace of this matrix (with eigenvalue 6). By adding it to the first matrix full of -1, it will then simply "translates" its spectrum by 6.

Thus eigenvalues of the original matrix are 6 (multiplicity 3) and 2 (multiplicity 1).