1. Tensor

 1.1. Definition

• An array containing multidimentional elements.
• N-order or N-way or N-mode tensor is an element of the tensor product of N vector spaces.

 1.2. Examples

• A first order tensor is a vector, and a second order tensor is a matrix.
• A Third order tensor isEach index is an element of $I$ vector space, $J$ vector space, $K$ vector space.

2. Notations

• The order of a tensor : The number of dimensions a.k.a. ways, mods
• Vectors : Boldface lowercase letters, e.g., a
  - The $i$th element of a vector a is $a_i$.
• Metrices : Boldface capital letters, e.g., A
  - An element $(i,j)$ of a matrix A is $a_{ij}$.
• Tensors : Boldface LaTex letters, e.g., $X$
  - An element $(i,j,k)$ of a tensor $X$ is denoted by $x_{ijk}$.
• The $n$th element of a sequence is denoted by a superscript in parentheses.
  e.g. A$^{(n)}$: $n$th matrix in a sequence
• Fibers : Fixing every index but one
  e.g. A matrix column : mode-1 fiber
       A matrix row : mode-2 fiber
       For a 3-order tensor:
• Slices : Fixing all but two indices, two-dimentional sections of a tensor.
• Norm : The norm of a tensor $X\in R^{I_1 \times I_2 \times \cdots \times I_N}$ ,
  denoted by $\parallel$$X$$\parallel = \sqrt{\displaystyle\sum_{i_1=1}^{I_1}\sum_{i_2=1}^{I_2}\cdots\sum_{i_n=1}^{I_N}x^2_{i_1i_2\cdots i_N}}$

2.1. Matricization: Transforming a Tensor into a Matrix

• a.k.a unfolding or flattening, is reordering elements of an N-way Tensor into a matrix.
• The mode-n matricization of a tensor $X\in R^{I_1 \times I_2 \times \cdots \times I_N}$ is denoted by $X_{(n)}$ and arranges the mode-n fibers to be the columns of the resulting matrix.
• Tensor element $(i_1,i_2,\dots,i_N)$ maps to matrix element $(i_n,j)$, where
  $j = 1+\displaystyle\sum_{\substack{k=1 \\ k\neq n}}^{N}(i_k-1)J_k$    with   $J_k = \displaystyle\prod_{\substack{m=1\\m\neq n}}^{k-1}I_m$

Describing mode-n fibers being the columns of the resulting matrix.