MathDB

n\times n

matrix

A

with integer entries is called representative if, for any integer vector

\mathbf{v}

, there is a finite sequence

0=\mathbf{v}_0,\mathbf{v}_1,\dots,\mathbf{v}_{\ell}=\mathbf{v}

of integer vectors such that for each

0\leq i <\ell

, either

\mathbf{v}_{i+1}=A\mathbf{v}_{i}

\mathbf{v}_{i+1}-\mathbf{v}_i

is an element of the standard basis (i.e. one of its entries is

1

, the rest are all equal to

0

). Show that

A

is not representative if and only if

A^T

has a real eigenvector with all non-negative entries and non-negative eigenvalue.

3