Basic operations
| Sum / difference | Element-wise; same shape required |
|---|---|
| Scalar multiplication | Each element × scalar |
| Matrix product A · B | (m×n)·(n×p) → (m×p); inner dims must match |
| Transpose Aᵀ | Swap rows and columns |
| Inverse A⁻¹ | A · A⁻¹ = I. Exists iff det(A) ≠ 0 |
| Identity I | Diagonal of 1s, elsewhere 0. I · A = A · I = A |
Properties
| Non-commutative | AB ≠ BA in general |
|---|---|
| Associative | (AB)C = A(BC) |
| Distributive | A(B + C) = AB + AC |
| Transpose rules | (A+B)ᵀ = Aᵀ + Bᵀ; (AB)ᵀ = BᵀAᵀ |
| Inverse rules | (AB)⁻¹ = B⁻¹ A⁻¹; (Aᵀ)⁻¹ = (A⁻¹)ᵀ |
Determinant
| 2×2 | det([[a,b],[c,d]]) = ad − bc |
|---|---|
| 3×3 | Sarrus or cofactor expansion |
| Properties | det(AB) = det(A)·det(B); det(Aᵀ) = det(A); det(A⁻¹) = 1/det(A) |
Special matrices
| Name | Property |
|---|---|
| Symmetric | A = Aᵀ |
| Skew-symmetric | A = −Aᵀ |
| Orthogonal | Aᵀ · A = I (Aᵀ = A⁻¹) |
| Diagonal | Non-zero only on main diagonal |
| Triangular (upper/lower) | Zero below/above diagonal |
| Unitary (complex) | A*·A = I |
| Positive definite | xᵀAx > 0 for all x ≠ 0 |
Eigen
| Eigenvalue eq | A·v = λ·v |
|---|---|
| Characteristic | det(A − λI) = 0 |
| Trace | Sum of diagonal = sum of eigenvalues |
| Det | Product of eigenvalues |
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