# Finite fast tests for periodicity of certain matrices

MathOverflow Asked on January 12, 2021

Let $$M=- U^{-1} U^T$$ be an $$n times n$$-integer matrix, where $$U$$ is an upper triangular 0-1-matrix where all diagonal entries are equal to one. $$M$$ is called periodic if $$M^r=id$$ for some $$r geq 1$$.
The question is about whether there is a fast finite test to check whether a matrix is periodic, so that one just has to look for "small" $$r$$. See Distributive lattices with periodic Coxeter matrix for a motivation.

Question 1: Given $$n$$, is there a good bound for the minimal number $$a(n)$$ such that if $$M^r neq id$$ for all $$r=1,…,a(n)$$, then $$M$$ is not periodic?

Question 2: Given $$n$$, is there a good bound for the minimal number $$b(n)$$ such that if $$M$$ has an entry of absolute value at least $$b(n)$$, then $$M$$ is not periodic?

Of course one might ask those questions also for integer matrices $$M$$ with more general properties. (there it does not work for question 2 as Gerry Myerson showed)

For general integer matrices $$M$$, there is no $$b(n)$$. E.g., for $$n=2$$, if $$a(a+1)+1=bc$$, and $$M=pmatrix{a&-bcr c&-a-1cr}$$ then $$M^3$$ is the identity matrix.

EDIT: for general matrices, Theorem 2.7 of James Kuzmanovich and Andrey Pavlichenkov, Finite groups of matrices whose entries are integers, The American Mathematical Monthly Vol. 109, No. 2 (Feb., 2002), pp. 173-186 shows there's a bound on $$a(n)$$ in terms of $$n$$. Let $$m=p_1^{e_1}p_2^{e_2}cdots p_t^{e_t}$$ with $$p_1. Then there is an $$ntimes n$$ integer matrix with order $$m$$ if and only if

1. $$sum_{i=1}^t(p_i-1)p_i^{e_i-1}-1le n$$ for $$p_1^{e_1}=2$$, or

2. $$sum_{i=1}^t(p_i-1)p_i^{e_i-1}le n$$ otherwise.

Answered by Gerry Myerson on January 12, 2021

## Related Questions

### Groupoids as models of symmetric simplicial sets

1  Asked on January 9, 2021 by ben-macadam

### Real part of a holomorphic section of a vector bundle

0  Asked on January 8, 2021 by user158773

### “Trade-off” between bound on the function and on the spectrum for functional calculus in spectral theory

0  Asked on January 8, 2021 by ma-joad

### Gaussian expectation of outer product divided by norm (check)

1  Asked on January 7, 2021 by b-merlot

### If $W$ is a Markov chain and $N$ is a Poisson process, then $left(W_{N_t}right)_{tge0}$ is Markov

0  Asked on January 7, 2021

### Open problems in matroid theory

2  Asked on January 6, 2021 by logictheorist

### Automorphism group of Hermitian symmetric spaces

1  Asked on January 5, 2021 by thiku

### Existence of Gaussian random field with prescribed covariance

1  Asked on January 5, 2021 by cabbage

### What, precisely, do we mean when we say that a f.d. vector space is canonically isomorphic to its double dual?

0  Asked on January 4, 2021 by qiaochu-yuan

### Discrete logarithm for polynomials

1  Asked on January 4, 2021 by adam-p-goucher

### Does the lemma remain valid in b-metric space?

0  Asked on January 4, 2021 by seddik-merdaci

### A determinant identity

1  Asked on January 4, 2021 by vassilis-papanicolaou

### Extending continuous functions from dense subsets to quasicompacts

1  Asked on January 3, 2021 by user97621

### Top and bottom composition factors of $M$ are isomorphic

1  Asked on January 2, 2021 by user666

### Widely accepted mathematical results that were later shown to be wrong?

47  Asked on January 2, 2021 by roman-starkov

### Anti-concentration of Gaussian quadratic form

3  Asked on January 1, 2021 by mitch

### Convergence of solving an OP $minlimits_{x, alpha} sum_i alpha_i f_i(x) + g(alpha)$

0  Asked on January 1, 2021 by user263322

### Quadrature for numerical integration over infinite intervals

1  Asked on December 31, 2020 by bernied

### When is ${b^2 – {b-1}_2}_2=1$ with odd $b$? (The bracket-notation explained below)

4  Asked on December 30, 2020 by gottfried-helms

### How to recognize a vector bundle?

2  Asked on December 30, 2020 by user163840

### Ask a Question

Get help from others!