Let
A
A
be a nonnegative real matrix which is expanding, i.e. with all eigenvalues
|
λ
|
>
1
|\lambda | > 1
, and suppose that
|
det
(
A
)
|
|\det (A)|
is an integer. Let
D
{\mathcal D}
consist of exactly
|
det
(
A
)
|
|\det (A)|
nonnegative vectors in
R
n
\mathbb {R}^n
. We classify all pairs
(
A
,
D
)
(A, {\mathcal D})
such that every
x
x
in the orthant
R
+
n
\mathbb {R}^n_+
has at least one radix expansion in base
A
A
using digits in
D
{\mathcal D}
. The matrix
A
A
must be a diagonal matrix times a permutation matrix. In addition
A
A
must be similar to an integer matrix, but need not be an integer matrix. In all cases the digit set
D
\mathcal D
can be diagonally scaled to lie in
Z
n
\mathbb {Z}^n
. The proofs generalize a method of Odlyzko, previously used to classify the one–dimensional case.