Matrix

A matrix (plural: matrices) is a 2-dimensional ordered list of numbers:

$$ \mathbf{M}_{m \times n} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}, $$ where element $a_{ij}$ is in row $i$, column $j$.

It can be therefore written:

$$ \mathbf{M}_{m \times n} = (a_{ij}),i \in \{1,\dots,m\}, j \in \{1,\dots,n\} $$

Formal definition

Any function satisfying

$$ \mathbf{M} : \{1,\dots,m\} \times \{1,\dots,n\} \to \mathbb{F} $$

is a matrix of dimension $m \times n$ over field $\mathbb{F}$.

Dimension

If a matrix $\mathbb{M}_{m \times n}$ has $m$ rows and $n$ columns, its dimension is $m \times n$.

Special cases:

	Known as
$m=n$	Square matrix
$m=1$	Row vector
$n=1$	Column vector

Binary operations

Single matrix operations

For matrice $\mathbf{A}=(a_{ij}) \in \mathbb{F}$:

Name	Operation
scalar multiplication	$c\mathbf{A} = (c\,a_{ij})$
transposition	$\mathbf{A}^T = (a_{ji})$

Two matrices operations

Matrix addition/subtraction

It is defined for two matrices of identical sizes $\mathbf{A}_{m \times n}=(a_{ij}), \mathbf{B}_{m \times n}=(b_{ij})\in \mathbb{F}$:

$$ (\mathbf{A} \pm \mathbf{B})= (a_{ij} \pm b_{ij}) $$

Matrix multiplication

It is defined for two matrices of compatible sizes $\mathbf{A}_{m \times n}=(a_{ij}), \mathbf{B}_{n \times p}=(b_{ij})\in \mathbb{F}$:

$$\begin{gathered} \forall i \in \{1,\dots,m\}, j \in \{1,\dots,p\} \\ (\mathbf{A}\mathbf{B})_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}. \end{gathered}$$

Notice that $\mathbf{AB}$ is a $m \times p$ matrix over $\mathbb{F}$

Identity matrix

If square matrix $\mathbf{I}=(a_{ij})$ of dimension $m \times m$ satisfies

$$ \begin{cases} i=j: a_{ij}=1 \\ i \neq j: a_{ij}=0 \end{cases} $$

then it is an identity matrix of dimension $m$, denoted $\mathbf{I}_m$.

From the definition of #Matrix multiplication, it can be inferred that

$$ \mathbf{A}_{m \times n}\mathbf{I}_n = \mathbf{A}_{m \times n} = \mathbf{I}_m\mathbf{A}_{m \times n} $$

Matrix operations in python

Initialisation

import numpy as np # import dependencies: always the 1st step!

The matrices are to be inputted as 2D numpy arrays, so:

$$ \forall a_{ij} \in \mathbf{A}, \text{a[i][j]}=a_{ij} $$

For example, to initialise the identity matrice $\mathbf{I}_3$:

$$ I=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix} $$

I=np.array([[1,0,0],[0,1,0],[0,0,1]])

For two matrices $\mathbf{A},\mathbf{B}$ initialised as 2D arrays:

Single matrix operations

Transposition, determinant, comatrix, inverse

A_T=A.T # Transpose A: -> 2D nparray: since A is an nparray, nupmy understands what you mean!
det_A=np.linalg.det(A): -> float
inverse_A=np.linalg.inv(A): -> 2D nparray

Element-wise,column-wise, and row-wise sum

sum_A=np.sum(A) # -> float: element-wise sum of A
column_wise_sum=np.sum(A, axis=0): -> 1D nparray
row_wise_sum=np.sum(A, axis=1): -> 1D nparray

Two matrices operations

Element-wise operations

sum_A_B=np.add(A,B) # A+B
difference_A_B=np.subtract(A,B) # A-B
element_wise_product_A_B=np.multiply(A,B) # A \circ B
quotient_A_B=np.divide(A,B) # A/B

Be extremely careful with np.multiply, which gives the element-wise product, NOT the matrix product.

Matrix operations

product_A_B=np.dot(A,B) # A.B