[Linear Algebra] Vector Matrix Multiplication

Vector / Matrix Additions and Multiplications

• Element-wise addition : $C = A + B$, $C_{ij} = A_{ij} + B_{ij}$
  • A, B, C should have the same size : $A, B, C \in \mathbb{R}^{m \times n}$
• Scalar multiple of vector/matrix : $c\textbf{a}$, $cA$
• Matrix-matrix multiplication : $C = AB$, $C_{ij} = \sum_{k} A_{i,k} B_{k,j}$
  • constraint : $A \in \mathbb{R}^{? \times n}, B \in \mathbb{R}^{n \times ?}$

Matrix Multiplication is NOT commutative

• Matrix multiplication is NOT commutative : $AB \neq BA$

Other Properties

• Distributive : $A(B+C) = AB + AC$
• Associative : $A(BC) = (AB)C$
• Property of transpose : $(AB)^T = B^T A^T$