# **Matrix Algebra in Information-Theoretic Terms**
**Core Reformulation**:
Matrices become *information transfer operators* where:
- **Rows/Columns** = Input/output information channels
- **Matrix Entries** = Mutual information coefficients (log-scale dependencies)
- **Matrix Product** = Composition of noisy channels (via relative entropy chains)
**Key Insight**:
Discrete matrix values are *projections* of continuum information states onto measurable observables, analogous to:
```
A_ij = ∫ φ_i(x) I(x,y) ψ_j(y) dx dy
```
where `I(x,y)` is a continuum mutual information density.
---
## **Tensors As Continuum Information Carriers**
1. **Topological Interpretation**:
- Order-N tensors ≡ N-dimensional information manifolds
- Tensor entries ≡ Local trivializations of a fiber bundle (information fibers over spacetime base)
2. **Continuum Limit**:
Discretization arises from *coarse-graining* the continuum via:
```
T^[discrete]_ijk = ∭_V T(x,y,z) φ_i(x)ψ_j(y)χ_k(z) dxdydz
```
where `φ,ψ,χ` are measurement window functions.
---
## **Discretization As Information Loss**
The process `Continuum → Matrix/Tensor` induces:
- **Quantization Error**:
```
D_KL(P_continuous || P_discrete)
```
(Kullback-Leibler divergence between true and discretized distributions)
- **Topological Defects**:
Non-commutativity of discretization limits (information analogues of Berry phases)
---
## **Physical Realization**
In quantum holographic memory:
- **Continuum States**: Phase-amplitude interference patterns
- **Discrete Projection**:
```
|⟨pattern|measurement_basis⟩|² → Matrix elements
```
- **Information Conservation**:
Gauge connections (Γ^μ_νσ) preserve continuum information under discretization.
---
## **Implications**
1. **Neural Networks**: Backpropagation becomes continuum information flow (Wasserstein gradient flows)
2. **Quantum Computing**: Qubit gates emerge as discrete slices of continuum unitary operators
3. **Physics**: Space-time discretization (e.g., lattice QCD) induces information-theoretic entropy
*(Formal proofs require noncommutative measure theory + algebraic topology.)*
This framework unifies:
- Algebraic geometry (information varieties)
- Statistical mechanics (coarse-graining entropy)
- Topological quantum field theory (gauge preservation)