Matrix Operations Cheat Sheet
Matrix arithmetic rules, transpose, inverse, determinant, and key properties.
Reference
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
Last updated: