Spectral Radius proof

Question

Let A be a symmetric matrix with positive Eigenvalues.Proof that ρ(Α^4)=(ρ(Α))^4 where ρ is the spectral radius.

How should i go about that?

I am kinda confused on how should i use the information i am given

Answer 1

The spectral radius formula is $\rho (A)=\lim \|A^{n}\|^{1/n}$. From this $\rho (A^{4})=\lim \|A^{4n}\|^{1/n}=(\lim \|A^{4n}\|^{1/4n})^{4}$ which is $\rho (A) ^{4}$. See Gelfand Formula in https://en.wikipedia.org/wiki/Spectral_radius