What operation on matrix which squares its eigenvalues but maintains its eigenvectors?

Question

I have a problem which requires me to use index ellipsoids. Specifically using the refractive index to find the speed of light in different directions. I was using the following formula. $$c\left(\hat r\right)=\frac{c_0}{\sqrt{{\hat{r}}^T\widetilde{n}\hat{r}}}$$ But I found that the speed of light index it gave me was wrong. While all the principal axes were in the right place, their magnitudes were the square root of what they should've been. While that formula above is good for getting ellipsoids, I need a different internal matrix. I've determined that the correct internal matrix should have the same eigenvectors but its eigenvalues should be squared. Does anyone know what operation I need to perform on $\widetilde{n}$ to get the desired matrix?

Answer 1

Let $M$ be a matrix that has some eigenvalues, then $M^2$, the square of that matrix, will have eigenvalues that are squares of old one.

$$ Mx =ax \implies M^2x = MMx = Max = aMx = a^2x $$