How to compute/estimate the eigenvalues of this matrix

Question

I'm trying to find the eigenvalues of a matrix $$A=\begin{bmatrix}2/3 & -1/4 & -1/4 \\ -1/4 & 2/3 & -1/4 \\ -1/4 & -1/4 & 2/3\end{bmatrix}$$

The eigenvalues of this matrix, are the roots $\lambda$ of the equation $det(A-\lambda I)=0$. Expanding this determinant with Sarrus's Rule gives a polynomial of a third degree, the solutions can apparently be estimated by iterative methods. Before I start exploring those avenues, however, I'd like to know if there is a more practical method to compute the eigenvalues of this matrix.

Answer 1

It's equal to $- \frac{1}{4}J + \frac{11}{12} I$, where $I$ is the identity matrix and $J$ is the matrix of all $1$s. Note that $J$ has two eigenvalues: $0$ with multiplicity $2$ (since it has a rank of $1$) and $3$ with multiplicity $1$, with eigenvector $(1, 1, 1)$.

So, the eigenvalues of $- \frac{1}{4}J$ are $-3/4, 0, 0$, and hence $- \frac{1}{4}J + \frac{11}{12} I$ has eigenvalues $1/6, 11/12, 11/12$.

Answer 2

Check for the rational roots of the characteristic polynomial.

The eigenvalues are $$\frac {11}{12}, \frac {11}{12}, \frac {1}{6} $$