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Matrix Product

Tool to calculate matrix product algebra. The matrix product consists of the multiplication of matrices.

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# Matrix Product

## Matrix Product

### Alphabet

Tool to calculate matrix product algebra. The matrix product consists of the multiplication of matrices.

### How to multiply 2 matrices?

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 $$M_1.M_2 \neq M_2.M_1$$

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

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

### 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.