How to find the eigenvalues of the following matrix?

Question

$$M=\left[\begin{array}{cc} 4I_k+4\left(\frac{m-2}{m+2}\right)J_{k,k} & -\frac{8}{m+2}J_{k,m-k} \\\ -\frac{8}{m+2}J_{m-k,k} & 4I_{m-k}-\frac{4}{m+2}J_{m-k,m-k}\\ \end{array}\right] $$

My calculated results do not match with the computer results. $I_m$ is an identity matrix and $J_{m,m}$ is a matrix of ones. Note that $2 \le k \le m-2 $.

Answer 1

Following Hans Engler, write $M = 4 I_m + B$, and let the nonzero eigenvalues of $B$ be $d_1$ and $d_2$.

Let $$\eqalign{t_1 &= \text{Trace}(B) = 4 k (m-2)/(m+2) - 4 (m-k)/(m+2)\cr t_2 &= \text{Trace}(B^2) = \dfrac{16}{(m+2)^2} \left(((m-2)^2 k+4 (m-k)) k + (4 k+m-k) (m-k)\right) \cr}$$ Then $t_1 = d_1 + d_2$ and $t_2 = d_1^2 + d_2^2$. Thus $d_1$ and $d_2$ are the roots of $ 2 x^2 - 2 t_1 x + t_1^2 - t_2 = 0$, i.e. $$d_1, d_2 = \dfrac{t_1}{2} \pm \dfrac{\sqrt{2 t_2 - t_1^2}}{2}$$

Answer 2

This is a rank 2 perturbation of a multiple of the identity matrix. It may be written in the form $M = 4I_m + B$ where $B$ is the remainder term. This term in turn has the form $B = d_1 aa^T + d_2 bb^T$ where $a$ and $b$ are orthogonal and have norm 1.

Then the eigenvalues are $4 + d_1, 4 + d_2$ (each with multiplicity 1) and 4 (multiplicity $m-2$).