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Matrix Product

Tool to calculate matrix products. Matrix product algebra consists of the multiplication of matrices (square or rectangular).

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# Matrix Product

## Matrix Product

### Alphabet

Tool to calculate matrix products. Matrix product algebra consists of the multiplication of matrices (square or rectangular).

### How to multiply 2 matrices?

Take $M_1=[a_{ij}]$ a matrix of $m$ lines and $n$ columns and $M_2=[b_{ij}]$ a matrix of $n$ lines and $p$ columns (2x2,2x3,3x2,3x3,etc.). 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 $$M_1.M_2 \neq M_2.M_1$$

### How to multiply a matrix by a scalar?

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

$$\lambda M = [ \lambda a_{ij} ]$$

### What are matrix multiplication properties?

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

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

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

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

### How to multiply 2 matrices of incompatible shapes?

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

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