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### Abstract

The spectral radius of a matrix A is the maximum norm of all
eigenvalues of A. In previous work we already formalized that for a
complex matrix A, the values in A^{n} grow polynomially in n
if and only if the spectral radius is at most one. One problem with
the above characterization is the determination of all
*complex* eigenvalues. In case A contains only non-negative
real values, a simplification is possible with the help of the
Perron–Frobenius theorem, which tells us that it suffices to consider only
the *real* eigenvalues of A, i.e., applying Sturm's method can
decide the polynomial growth of A^{n}.

We formalize
the Perron–Frobenius theorem based on a proof via Brouwer's fixpoint
theorem, which is available in the HOL multivariate analysis (HMA)
library. Since the results on the spectral radius is based on matrices
in the Jordan normal form (JNF) library, we further develop a
connection which allows us to easily transfer theorems between HMA and
JNF. With this connection we derive the combined result: if A is a
non-negative real matrix, and no real eigenvalue of A is strictly
larger than one, then A^{n} is polynomially bounded in n.

### Change history

#### May 17, 2018

prove conjecture of CPP'18 paper: Jordan blocks of spectral radius have maximum size (revision ffdb3794e5d5)

#### October 18, 2017

added Perron-Frobenius theorem for irreducible matrices with generalization (revision bda1f1ce8a1c)### Topics

### Theories of Perron_Frobenius

- Cancel_Card_Constraint
- Bij_Nat
- HMA_Connect
- Perron_Frobenius_Aux
- Perron_Frobenius
- Roots_Unity
- Perron_Frobenius_Irreducible
- Perron_Frobenius_General
- Spectral_Radius_Theory
- Spectral_Radius_Largest_Jordan_Block
- Hom_Gauss_Jordan
- Spectral_Radius_Theory_2
- Check_Matrix_Growth