Tensor

A tensor is a mathematical object that can represent data in different dimensions

Dimensions	Name	Description	Notation
0D	Scalar	A single number	$k$
1D	Vector	A list of numbers	$\mathbf{v}=\left[\begin{array}{cccc}{v}_1& v_2& v_3& \dots \end{array}\right]$
2D	Matrix	A grid of numbers	$\mathbf{A}=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1,y}\\ a_{21}& a_{22}& \dots & a_{2,y}\\ ⋮& ⋮& \ddots & ⋮\\ a_{x,1}& a_{x,2}& \dots & a_{x,y}\end{array}\right]$
3D+	High-dimensional array	–	$\mathbf{T}=\left[\begin{array}{cc}\left[\begin{array}{cc}{t}_{111}& t_{112}\\ t_{121}& t_{122}\end{array}\right],\left[\begin{array}{cc}{t}_{211}& t_{212}\\ t_{221}& t_{222}\end{array}\right],& \dots \end{array}\right]$

Example of applications

• Algebra: to apply operations on data with multiple dimensions
• Physics: to describe physical properties (stress, strain, electromagnetic fields, spacetime, …)
• Machine learning: to store and manipulate data (c.f. tensor in machine learning)