Sparse Matrix Decompositions¶

The algebrax.matrix.decompose module provides factorizations and matrix decompositions for sparse dictionary matrices.

Supported Decompositions¶

Decomposition	Function	Invariant	Description
LU (Pivoted)	`lu(A)`	\(P \cdot A = L \cdot U\)	Unit lower-triangular \(L\), upper-triangular \(U\), permutation matrix \(P\).
QR (Gram-Schmidt)	`qr(A)`	\(A = Q \cdot R\)	Orthonormal column matrix \(Q\) (\(Q^T Q = I\)), upper-triangular \(R\).
SVD (Truncated)	`svd(A, k=...)`	\(A \approx U \cdot \text{diag}(S) \cdot V^T\)	Singular vectors \(U, V^T\) and singular values vector \(S\).
Cholesky	`cholesky(A)`	\(A = L \cdot L^T\)	Lower-triangular \(L\) for symmetric positive-definite matrices.

Python Example¶

import algebrax as ax

# 1. Cholesky Decomposition
A_spd = {
    "0": {"0": 4.0, "1": 12.0},
    "1": {"0": 12.0, "1": 37.0},
}
L = ax.matrix.cholesky(A_spd)
print("Cholesky L:", L)
print("L @ L^T == A:", ax.matrix.dot(L, ax.matrix.transpose(L)) == A_spd)

# 2. LU Decomposition
A = {
    "0": {"0": 1.0, "1": 2.0, "2": 4.0},
    "1": {"0": 3.0, "1": 8.0, "2": 14.0},
    "2": {"0": 2.0, "1": 6.0, "2": 13.0},
}
P, L, U = ax.matrix.lu(A)
print("P @ A == L @ U:", ax.matrix.dot(P, A) == ax.matrix.dot(L, U))

# 3. QR Decomposition
Q, R = ax.matrix.qr(A)
print("Q @ R == A:", ax.matrix.dot(Q, R) == A)

# 4. Truncated SVD (Top k=1 singular component)
U1, S1, VT1 = ax.matrix.svd(A, k=1)
print("Top Singular Value:", S1[0])