MathDB

For an invertible

n\times n

matrix

M

with integer entries we define a sequence

\mathcal{S}_M=\{M_i\}_{i=0}^{\infty}

by the recurrence

M_0=M

M_{i+1}=(M_i^T)^{-1}M_i

for

i\geq 0

.
Find the smallest integer

n\geq 2

for wich there exists a normal

n\times n

matrix with integer entries such that its sequence

\mathcal{S}_M

is not constant and has period

P=7

i.e

M_{i+7}=M_i

. (

M^T

means the transpose of a matrix

M

. A square matrix is called normal if

M^T M=M M^T

holds).
Proposed by Martin Niepel (Comenius University, Bratislava)..

Problem Statement