# counter example of two matrices with different number of positive and negative eigen values satisfying given constraint

consider following $$2$$ matrices of $$3\times3$$ order with all entries from real numbers $$\,\,$$ $$\begin{pmatrix} a & b & c \\ p & q & r\\ x & y & z \\ \end{pmatrix}$$ and $$\begin{pmatrix} a & \frac{b+p}{2} & \frac{c+x}{2} \\ \frac{b+p}{2} & q & \frac{r+y}{2}\\ \frac{c+x}{2} & \frac{r+y}{2} & z \\ \end{pmatrix}$$ I want to find a counter example such that both matrices have different number of positive and negative eigenvalues. I was unable to find a counter example of it. I thought to associate this question with quadratic form and signature of quadratic form but i got stuck how to proceed further . Thank you

• Is it your intention that neither matrix should have an eigenvalue zero? Or that all eigenvalues (of the first matrix) are necessarily real? Apr 3, 2020 at 16:13
• both matrices can have eigenvalue 0. Apr 3, 2020 at 16:21
• Must all of the eigenvalues be real?
– amd
Apr 3, 2020 at 20:40

Set $$a=b=c=q=r=1$$ and all the others to $$0$$. First matrix has eigenvalues $$1$$ and $$0$$, while the second has eigenvalues $$-0.28, 0.5, 1.78$$.
Note: Denoting the first matrix by $$A$$, the second can be seen to be $$\frac{1}{2}(A+A^T)$$.