Eigenvalues of matrix subtraction

Question

Given the matrix A:

\begin{equation*} A = \begin{pmatrix} 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\ 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\ 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\ 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\ -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 \\ -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 \\ -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 \\ -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 \\ \end{pmatrix} \end{equation*}

I know that its eigenvalues are 0 (with Multiplicity of 7) and 8. How can I easily find the eigenvalues of the Matrix: T = 0.5I-$\mu$A? What is the relation between the original eigenvalues of A and 0.5I-$\mu$A?

Answer 1

The eigenvalues of $\alpha A+\beta I$ are $\alpha\lambda+\beta$ where $\lambda$ is an eigenvalue of $A$.