An n×n matrix is a linear operator on kⁿ once a basis is fixed. Determinant det(A) = Σ_σ sgn(σ) ∏ a_{i,σ(i)} detects invertibility (A invertible ⇔ det A ≠ 0).
Matrix and determinant
Related concepts
- Linear map
- Change of basis
- Singular value decomposition (SVD)
- Condition number
- Gaussian elimination
- Cayley–Hamilton theorem
- Quadratic form
- Trace tr(A)
- so(n) orthogonal Lie algebra: antisymmetric n×n matrices
- Jordan algebra (commutative nonassociative with Jordan identity)
- Sym_2(R) Jordan product: commutativity and Jordan identity literal-0 residuals
- Jordan power-associativity: (X²)∘X = X∘(X²) = X³ on Sym_2(R)
- Legendre condition (second variation)
- Atiyah-Singer index theorem
- Zorn vector matrix (octonion realisation)
- Spectral sequences (Leray-Serre)
- Character theory (Frobenius)
- Fisher information + Cramer-Rao bound
- Compressed sensing (Candès-Tao-Donoho)
- Polar codes (Arıkan)