# Determine all possible real a, so that matrix ((1,a,a),(a,1,a),(a,a,1)) has 3 positive real eigen value ?

Apr 10, 2017

See below.

#### Explanation:

Giving $M = \left(\begin{matrix}1 & a & a \\ a & 1 & a \\ a & a & 1\end{matrix}\right)$ it's characteristic polynomial is

$p \left(\lambda\right) = {\lambda}^{3} - 3 {\lambda}^{2} + 3 \left(1 - {a}^{2}\right) \lambda - 2 {a}^{3} + 3 {a}^{2} - 1 =$

${\left(\lambda + a - 1\right)}^{2} \left(1 + 2 a - \lambda\right)$

so this polynomial has three real roots or

$p \left(\lambda\right) = {\left(\lambda + a - 1\right)}^{2} \left(1 + 2 a - \lambda\right)$

with eigenvalues

$\left(1 - a , 1 - a , 1 + 2 a\right)$.

This result does not surprise us because $M$ is hermitian and a hermitian matrix has always real eigenvalues.

NOTE:
Positive eigenvalues are obtained for $1 - a > 0$ and $1 + 2 a > 0$ or

$- \frac{1}{2} < a < 1$