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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

## Alphabet

### What is a matrix product? (Definition)

The matrix product is the name given to the most common matrix multiplication method.

$M_1=[a_{ij}]$ is a matrix of $m$ lines and $n$ columns and $M_2=[b_{ij}]$ is a matrix of $n$ lines and $p$ columns (all formats are possible 2x2, 2x3, 3x2, 3x3, 3x4, 4x3, etc.). The matrix product $M_1.M_2 = [c_{ij}]$ is a matrix of $m$ lines and $p$ columns, with: $$\forall i, j : c_{ij} = \sum_{k=1}^n a_{ik}b_{kj}$$

The multiplication of 2 matrices $M_1$ and $M_2$ is noted with a point $\cdot$ or . so $M_1 \cdot M_2$

The matrix product is only defined when the number of columns of $M_1$ is equal to the number of rows of $M_2$ (matrices are called compatible)

### How to multiply 2 matrices? (Matrix product)

The multiplication of 2 matrices $M_1$ and $M_2$ forms a result matrix $M_3$. The matrix product consists in carrying out additions and multiplications according to the positions of the elements in the matrices $M_1$ and $M_2$.

$$M_1 = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \\ M_2 = \begin{bmatrix} b_{11} & b_{12} & \cdots & b_{1p} \\ b_{21} & b_{22} & \cdots & b_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ b_{n1} & b_{n2} & \cdots & b_{np} \end{bmatrix} \\ M_1 \cdot M_2 = \begin{bmatrix} a_{11}b_{11} +\cdots + a_{1n}b_{n1} & a_{11}b_{12} +\cdots + a_{1n}b_{n2} & \cdots & a_{11}b_{1p} +\cdots + a_{1n}b_{np} \\ a_{21}b_{11} +\cdots + a_{2n}b_{n1} & a_{21}b_{12} +\cdots + a_{2n}b_{n2} & \cdots & a_{21}b_{1p} +\cdots + a_{2n}b_{np} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1}b_{11} +\cdots + a_{mn}b_{n1} & a_{m1}b_{12} +\cdots + a_{mn}b_{n2} & \cdots & a_{m1}b_{1p} +\cdots + a_{mn}b_{np} \end{bmatrix}$$

To calculate the value of the element of the matrix $M_3$ in position $i$ and column $j$, extract the row $i$ from the matrix $M_1$ and the row $j$ from the matrix $M_2$ and calculate their dot product. That is, multiply the first element of row $i$ of $M_1$ by the first element of column $j$ of $M_2$, then the second element of row $i$ of $M_1$ by the second element of the column $j$ of $M_2$, and so on, note the sum of the multiplications obtained, it is the value of the scalar product, therefore of the element in position $i$ and column $j$ in $M_3$.

Example: $$\begin{bmatrix} 1 & 0 \\ -2 & 3 \end{bmatrix} \cdot \begin{bmatrix} 2 & -1 \\ 4 & -3 \end{bmatrix} = \begin{bmatrix} 1 \times 2 + 0 \times 4 & 1 \times -1 + 0 \times -3 \\ -2 \times 2 + 4 \times 3 & -2 \times -1 + 3 \times -3 \end{bmatrix} = \begin{bmatrix} 2 & -1 \\ 8 & -7 \end{bmatrix}$$

### 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 as the initial matrix $M$, with each item 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: $$A \times (B + C) = A \times B + A \times C$$

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

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

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

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

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

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