Advanced Matrix Invariants & Academic Operations¶

The algebrax.matrix.academic module provides classical matrix invariants for square sparse dictionary matrices.

Warning

Exact academic functions like determinant(), cofactor(), and adjoint() involve recursive Laplace expansion or \(O (N^5)\) symbolic expansions. They emit a PerformanceWarning for \(N > 10\) and are designed for academic demonstration and exact symbolic verification.

Academic Functions¶

Function	Operation	Description
`adjoint(matrix)`	\(\text{adj}(A)\)	Classical adjugate matrix (transpose of cofactor matrix).
`determinant(matrix)`	\(\det(A)\)	Recursive Laplace expansion determinant of square sparse matrix.
`cofactor(matrix)`	\(C_{i,j}\)	Matrix of cofactors \(C_{i,j} = (-1)^{i+j} \det(M_{i,j})\).
`inverse(matrix)`	\(A^{-1}\)	Matrix inverse computed via adjugate division \(A^{-1} = \frac{1}{\det(A)} \text{adj}(A)\).

[!NOTE] For graph spectral analysis and node ranking (eigen_centrality, pagerank, fiedler_vector), see Discrete Calculus & Spectral Graph Theory.

Code Example¶

import algebrax as ax

# 1. Square 3x3 Sparse Matrix
A = {
    "0": {"0": 1.0, "1": 2.0, "2": 3.0},
    "1": {"0": 0.0, "1": 1.0, "2": 4.0},
    "2": {"0": 5.0, "1": 6.0, "2": 0.0},
}

# 2. Determinant & Inverse
det_A = ax.matrix.academic.determinant(A)
inv_A = ax.matrix.academic.inverse(A)
print(f"det(A) = {det_A}")
print("A^-1:", inv_A)

# 3. Cofactor & Adjugate
C = ax.matrix.academic.cofactor(A)
adj_A = ax.matrix.academic.adjoint(A)
print("Cofactor Matrix:", C)
print("Adjugate Matrix:", adj_A)