Prove spectral radius of a primitive matrix is 1

Asked May 24 '20 at 15:49

Active May 24 '20 at 16:05

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Let $P \in M (n \times n, \mathbb{R})$ be a primitive matrix. $1$ is a eigenvalue of $P$ and $(1,\dots,1)$ is the associated right eigenvector.

How can show that the spectral radius $\rho(P):=$max$\{|\lambda| : \lambda$ is a eigenvalue of $P\}$ of $P$ is 1?

Hint: Use Gelfand's formula.

edited May 24 '20 at 16:05

asked May 24 '20 at 15:49

Tino

a primitive matrix is stochastic, and alla stochastic matrices have 1 as spectral radius – Exodd May 24 '20 at 15:58
@Exodd Unfortunely no. – Tino May 24 '20 at 16:05
Why not? $P$ is stochastic, and in the link is proved that the spectral radius of a stochastic matrix is 1 – Exodd May 24 '20 at 16:09
1

How can I use Gelfand's formula to prove it? It is not included in the link. – Tino May 24 '20 at 16:11

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