---
title: 'Digit Matrices: Finite Expansions & Algorithms'
url: https://www.emergentmind.com/topics/digit-matrices
type: topic
---

# Digit Matrices: Finite Expansions & Algorithms

A digit matrix is an integer or rational matrix that serves as the “base” in a higher-dimensional positional representation system, generalizing classical scalar base expansions to objects in $\mathbb{Z}^n$ or more generally, $\mathbb{Z}^n[A]$. The associated machinery enables finite "radix" expansions of lattice vectors analogous to positional number systems, utilizing a finite digit set as representatives. This theory unifies ideas from algebraic number systems, symbolic dynamics, and tiling theory, and allows for explicit, algorithmic constructions of expansions in both expanding and non-expanding cases.

## 1. Fundamental Definitions and Finiteness Characterization

Let $A \in M_n(\mathbb{Q})$ (or $M_n(\mathbb{Z})$ in the integral case). The minimal $A$-invariant $\mathbb{Z}$-module containing $\mathbb{Z}^n$ is defined as
$$
\mathbb{Z}^n[A] = \bigcup_{k=1}^\infty \big(\mathbb{Z}^n + A\mathbb{Z}^n + \cdots + A^{k-1}\mathbb{Z}^n\big).
$$
A *digit system* is a pair $(A, \mathcal{D})$ with $\mathcal{D}\subset\mathbb{Z}^n[A]$ finite. The system is said to have the *finiteness property* if every $z \in \mathbb{Z}^n[A]$ admits a finite expansion of the form
$$
z = d_0 + A d_1 + \cdots + A^{k} d_{k},\qquad d_j \in \mathcal{D}.
$$
The main criterion is: **$(A, \mathcal{D})$ has the finiteness property for some finite $\mathcal{D}$ if and only if $A$ has no eigenvalue with absolute value $< 1$.** In such a case, $\mathcal{D}$ can be taken in $\mathbb{Z}^n$ itself [1801.01839], [2107.14168].

## 2. Algorithmic Construction and Explicit Digit Sets

The explicit construction of suitable digit sets involves canonical forms and convex geometry:

1. **Block decomposition:** Bring $A$ to rational or real (hyper-)companion form, decomposing into blocks associated to irreducible factors of its characteristic polynomial.
2. **Polynomial analogue:** For a block corresponding to $P(x)^{m}$, a finite set $N\subset\mathbb{Z}$ exists with $\mathbb{Z}[x]=N[x]+(P)$ iff all roots of $P$ satisfy $|\alpha|\ge1$.
3. **Residue systems:** Construct digit sets as coset representatives for $\mathbb{Z}^n[A]/A\mathbb{Z}^n[A]$. In expanding cases, any such system yields finiteness [2507.06158], [2107.14168].
4. **Convex hull conditions:** For unimodular rotation blocks ($|\lambda|=1$), convex-geometric requirements on digits ensure all "angles" are covered so the algorithm converges to 0 [2107.14168].
5. **Block sum and twisting:** Combine individual digit sets for each block by direct sum and apply necessary conjugations.
6. **Optimization:** Redundant digits may be pruned by analyzing the "repeller" set induced by the remainder-division map and using integer programming [2107.14168].

A typical algorithm proceeds by: (i) block-decomposing $A$; (ii) computing Smith normal forms for residue classes; (iii) selecting/or optimizing digit representatives by convex-hull or contraction criteria; and (iv) constructing the global digit system.

## 3. Dynamical Systems Perspective and Attractors

Digit matrix systems can be interpreted via discrete dynamical systems. For a digit set $\mathcal{D}$ providing complete residue classes modulo $A\mathbb{Z}^n[A]$, define the digit function $\delta(x)$ and backward map (remainder-division or beta-map)
$$
\Phi(x) := A^{-1}(x - \delta(x)).
$$
If $A$ is expanding, $\Phi$ has a *finite attractor* $\mathcal{A}_\Phi$ that eventually absorbs all orbits. Each $x\in \mathbb{Z}^n[A]$ admits a unique representation
$$
x = d_0 + A d_1 + \cdots + A^{k} d_{k} + A^{k+1}p, \quad p \in \mathcal{A}_\Phi.
$$
Finiteness property corresponds to $\mathcal{A}_\Phi = \{0\}$. Otherwise, the attractor describes inherent periodic structures in the digit system, generalizing purely periodic expansions in non-integer bases [2507.06158], [2107.14168].

## 4. Expanding, Non-Expanding, and Contractive Cases

| Case            | Defining Feature                      | Digit System Behavior                          |
|-----------------|--------------------------------------|------------------------------------------------|
| Expanding       | All $|\lambda|>1$                    | All vectors admit finite radix expansions; unique representations for each $x$ [2507.06158], [1002.4016] |
| Unimodular      | All $|\lambda|=1$                    | Finiteness only with appropriately “balanced” or convex-enclosed digit sets; possible lack of uniqueness [2107.14168], [2103.02599] |
| Non-Expanding   | Some $|\lambda|<1$                   | No digit system with finiteness property; expansions cannot reach all elements [1801.01839]              |
| Non-expansive Jordan block | $A \sim J_n(1)$         | Expansions always possible with at most $n$ digits; full systems characterized by parity, residue, or block combinatorics [2110.11937] |

This categorization clarifies why the spectral criterion $|\lambda|\ge 1$ is both necessary and sufficient for the finiteness property, whereas classic Euclidean division and its multidimensional analogues fail for matrices with contractive directions.

## 5. Notable Special Cases and Examples

### Expanding Matrices
Diagonal $A = \mathrm{diag}(b, ..., b), b \geq 2$: Coordinate-wise digit sets $\{0, 1, ..., b-1\}^n$ yield classical base-$b$ expansion in $\mathbb{Z}^n$ [1002.4016].

### Rational Rotations
$A = \frac{1}{5}\begin{pmatrix}3 & -4\\ 4 & 3\end{pmatrix}$: A rotation in $\mathbb{Z}^2$ admits a digit set constructed by residue-system and convex-enclosure methods; the resulting system is finite after refining via repeller sets [2107.14168].

### Companion Matrices of Polynomials
For quadratic $p(x) = x^2+\alpha x + \beta \in \mathbb{Q}[x]$, with $A = \begin{pmatrix}0 & -\beta\\ 1 & -\alpha\end{pmatrix}$, digits are coset representatives modulo $b = \mathrm{const}$. The system is finite if $0<\alpha\le\beta-1$. Attractors and automata describe all possible expansions and carries [2507.06158].

### Non-expansive Jordan Blocks
$J_n(1)$: For $n=2$, full expansion is characterized by explicit parity and gcd conditions on digits. For $n\ge3$, three digits suffice—canonical choices are $(0,\dots,0,\pm1)^T$, $(0,\dots,0,0)^T$; for similar matrices at most $n$ digits are necessary [2110.11937].

## 6. Generalizations and Connections

The digit matrix approach is a multidimensional generalization of classical positional numeration systems. It relates deeply to:

- The *height-reducing property* in algebraic number theory and digit representations in number fields [1801.01839].
- Euclidean division algorithms generalized to modules and lattices [1002.4016].
- Symbolic dynamics: representation languages of zero are in general not context-free, but can be recognized in $\mathsf{LSPACE}$ by logarithmic-memory Turing machines [2110.11937].
- Algebraic tiling and self-affine tiles: attractors for higher-rank systems yield fractal boundaries and nontrivial topological features [2507.06158].

## 7. Quantitative and Algorithmic Aspects

Quantitative bounds for expansion lengths and digit set sizes follow from block structure and polynomial bounds:
- For a companion matrix of $P(x) = a_d x^d + \cdots + a_0$, digit set cardinality can be $O(|a_0| + |a_d|)$.
- For matrices with spectral radius $\rho(A)>1$, expansion length is bounded above by $C \frac{\log \|z\|}{\log \rho(A)}$ for a suitable norm and constant $C$ [1801.01839].
- Construction algorithms rely on Smith normal form, convex-geometry (for unimodular rotations), and integer programming for minimality [2107.14168].

The interplay of spectral properties, lattice theory, and symbolic encoding enables systematic algorithmic construction and minimality improvement, integrating classical number-theoretic ideas with modern computational algebra and dynamics.

Source: https://www.emergentmind.com/topics/digit-matrices