Eigenvalues of two symmetric $4\times 4$ matrices: why is one negative of the other?

Question

Consider the following symmetric matrix:

$$ M_0 = \begin{pmatrix} 0 & 1 & 2 & 0 \\ 1 & 0 & 4 & 3 \\ 2 & 4 & 0 & 1 \\ 0 & 3 & 1 & 0 \end{pmatrix} $$

and a very similar matrix:

$$ M_1 = \begin{pmatrix} 0 & 1 & 2 & 0 \\ 1 & 0 & -4 & 3 \\ 2 & -4 & 0 & 1 \\ 0 & 3 & 1 & 0 \end{pmatrix} $$

To my surprise, the eigenspectrum of $M_0$ and $(-M_1)$ are the same! Why would this be the case?

I also tried playing around with the values a little; for example, if the center block is $\begin{pmatrix}1 & \pm 4 \\ \pm 4 & 1\end{pmatrix}$ instead, then they do not share the same eigenvalues.

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Context: I was considering the Hermitian matrix of this form ($M_2$ below) and noted that this has the same property as the matrix $M_0$ from above. Thus, presumably, it has nothing to do with the fact that the middle block is complex.

$$ M_2 = \begin{pmatrix} 0 & 1 & 2 & 0 \\ 1 & 0 & e^{ix} & 3 \\ 2 & e^{-ix} & 0 & 1 \\ 0 & 3 & 1 & 0 \end{pmatrix} $$

ps. I will accept any answer which explains the phenomenon between the real matrices. I think that would give a hint as to why $M_2$ / Hermitian matrices have the same property.

Thanks.

Answer 1

$$-M_1=D^{-1}M_0D$$ where $D=D^{-1}$ is the diagonal matrix with diagonal entries $(-1,1,1,-1)$. Therefore $M_0$ and $-M_1$ are conjugate, and have the same spectrum. This works because of the zeroes in the corners of $M_0$. In general, $$\pmatrix{a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ a_{41}&a_{42}&a_{43}&a_{44}}$$ and $$-\pmatrix{-a_{11}&a_{12}&a_{13}&-a_{14}\\ a_{21}&-a_{22}&-a_{23}&a_{24}\\ a_{31}&-a_{32}&-a_{33}&a_{34}\\ -a_{41}&a_{42}&a_{43}&-a_{44}}$$ are conjugate, for precisely the same reason.

Answer 2

I'm not sure if what follows is the type of thing you're looking for, but maybe you'll find this useful.

Consider the matrix $$ M_a = \left[\begin{array}{rrrr} 0 & 1 & 2 & 0 \\ 1 & 0 & a & 3 \\ 2 & a & 0 & 1 \\ 0 & 3 & 1 & 0 \end{array}\right] $$ The characteristic polynomials of $M_a$ and $M_{-a}$ are \begin{align*} \chi_{M_a}(t) &= t^{4} - \left(a^{2} + 15\right) t^{2} - 10 \, a t + 25 \\ \chi_{M_{-a}}(t) &= t^{4} - \left(a^{2} + 15\right) t^{2} + 10 \, a t + 25 \end{align*} Now, note that $\lambda$ is an eigenvalue of $M_a$ if and only if \begin{align*} 0 &= \chi_{M_a}(t) \\ &= {\lambda}^{4} - {\left(a^{2} + 15\right)} {\lambda}^{2} - 10 \, a {\lambda} + 25\\ &= (-\lambda)^{4} - {\left(a^{2} + 15\right)} (-\lambda)^{2} + 10 \, a (-\lambda) + 25 \\ &= \chi_{M_{-a}}(-\lambda) \end{align*} This proves that $M_{a}$ and $M_{-a}$ have eigenvalues related by negation.

Now, suppose that $M$ instead takes the form $$ M_{a+bi}=\left[\begin{array}{rrrr} 0 & 1 & 2 & 0 \\ 1 & 0 & a + i \, b & 3 \\ 2 & a - i \, b & 0 & 1 \\ 0 & 3 & 1 & 0 \end{array}\right] $$ In this case, the characteristic polynomials of $M_{a+bi}$ and $M_{-a+bi}$ are \begin{align*} \chi_{M_{a+bi}}(t) &= t^{4} + \left(-a^{2} - b^{2} - 15\right) t^{2} - 10 \, a t + 25 \\ \chi_{M_{-a+bi}}(t) &= t^{4} + \left(-a^{2} - b^{2} - 15\right) t^{2} + 10 \, a t + 25 \end{align*} A similiar argument then shows that $M_{a+bi}$ and $M_{-a+bi}$ have eigenvalues related by negation.

Answer 3

This is happening because of the somewhat special pattern of zeroes in this matrix. Edit: No it's not. It has everything to do with signature matrices instead, as shown in the other answer.

Let $$M_1 = \begin{bmatrix}0 & a_2 & a_3 & 0\\b_1 & 0 & b_3 & b_4\\c_1 & c_2 & 0 & c_4\\0 & d_2 & d_3 & 0\end{bmatrix}, \quad M_2 = \begin{bmatrix}0 & a_2 & a_3 & 0\\b_1 & 0 & -b_3 & b_4\\c_1 & -c_2 & 0 & c_4\\0 & d_2 & d_3 & 0\end{bmatrix}$$

Let $(\lambda, x)$ be an eigenvalue-eigenvector pair of $M_1$, where $x = \begin{bmatrix}x_1 & x_2 & x_3 & x_4\end{bmatrix}^T$. Then we can show that $\begin{bmatrix}x_1 & -x_2 & -x_3 & x_4\end{bmatrix}^T$ is an eigenvector corresponding to eigenvalue $-\lambda$ for $M_2$.

For, \begin{align*} a_2 x_2 + a_3 x_3 = \lambda x_1 & \implies a_2 (-x_2) + a_3(-x_3) = -\lambda x_1\\ b_1 x_1 + b_3 x_3 + b_4 x_4 = \lambda x_2 & \implies b_1 x_1 - b_3(-x_3) + b_4x_4 = (-\lambda)(-x_2). \end{align*} And the cases of the third and fourth rows are obviously similar.