Matrix Product

Consider \( M_1=[a_{ij}] \) a matrix of \( m \) lines and \( n \) columns and \( M_2=[b_{ij}] \) a matrix of \( n \) lines and \( p \) columns. The matrix product \( M_1.M_2 = [c_{ij}] \) is a matrix of \( m \) lines and \( p \) columns, with: $$ \forall i, j : c_{ij} = \sum_{k=1}^n a_{ik}b_{kj} $$

Example: $$ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 \times 1 + 2 \times 0 & 1 \times 0 + 2 \times 1 \\ 3 \times 1 + 4 \times 0 & 3 \times 0 + 4 \times 1 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $$

The order of the operands matters with matrix computations, so $$ M_1.M_2 \neq M_2.M_1 $$

The product of the matrix \( M=[a_{ij}] \) by a scalar \( \lambda \) is a matrix of the same size than the initial matrix M, with each items of the matrix multiplied by \( \lambda \). $$ \lambda M = [ \lambda a_{ij} ] $$

Associativity : $$ A \times (B \times C) = (A \times B) \times C $$

Distributivity : $$ A \times (B + C) = A \times B + A \times C $$

$$ (A + B) \times C = A \times C + B \times C $$

$$ \lambda (A \times B) = (\lambda A) \times B = A \times (\lambda B) $$

There is a matrix product compatible with any matrix sizes: the Kronecker product.