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

Question

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

Answer 1

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 $\frac{1}{2}(A+A^T)$.