# Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification
## 3 Tensors and Multilinear Forms  
In signal processing applications, a tensor \(\mathcal{X}\in\mathbb{K}^{I_{1}\times\cdots\times I_{N}}\) of order \(N\) and size \(I_{1}\times\cdots\times I_{N}\) is typically viewed as an array of numbers \([x_{i_{1},\cdots,i_{N}}]\). The order corresponds to the number of indices \((i_{1},\cdots,i_{N})\) that characterize its elements \(x_{i_{1},\cdots,i_{N}}\in\mathbb{K}\), also denoted \(x_{i_{1},\cdots,i_{N}}\) or \((\mathcal{X})_{i_{1},\cdots,i_{N}}\). Each index \(i_{n}\in\langle I_{n}\rangle\triangleq\{1,\cdots,I_{n}\}\), for \(n\in\langle N\rangle\triangleq\{1,\cdots,N\}\), is associated with a mode, also called a way, and \(I_{n}\) denotes the dimension of the \(n\)th mode. The number of elements in \(\mathcal{X}\) is equal to \(\prod_{n=1}^{N}I_{n}\). For instance, in a wireless communication system [8], each index of a signal \(x_{i_{1},\cdots,i_{N}}\) corresponds to a different form of diversity (in time, space, frequency, code, etc., domains), and the dimensions \(I_{n}\) are the numbers of time samples, receive antennas, subcarriers, the code length, etc. The tensor \(\mathcal{X}\) is said to be real (resp. complex) if its elements are real numbers (resp. complex numbers), which corresponds to \(\mathbb{K}=\mathbb{R}\) (resp. \(\mathbb{K}=\mathbb{C}\)). It is said to have even order (resp. odd order) if \(N\) is even (resp. odd). The special cases \(N=2\) and \(N=1\) correspond to the sets of matrices \(\mathbf{X}\in\mathbb{K}^{I\times I}\) and column vectors \(\mathbf{x}\in\mathbb{K}^{I}\), respectively. If \(I_{1}=\cdots=I_{N}=I\), the \(N\)th-order tensor \(\mathcal{X}=[x_{i_{1},\cdots,i_{N}}]\in\mathbb{K}^{I\times I\times\cdots\times I}\) is said to be hypercubic, of dimensions \(I\), with \(i_{n}\in\langle I\rangle\), for \(n\in\langle N\rangle\). The number of elements in \(\mathcal{X}\) is then equal to \(I^{N}\). The set of (real or complex) hypercubic tensors of order \(N\) and dimensions \(I\) will be denoted \(\mathbb{K}^{[N;I]}\). A hypercubic tensor of order \(N\) and dimensions \(I\) is said to be symmetric if it is invariant under any permutation \(\pi\) of its modes, i.e.,  
\[a_{\pi(i_{1},i_{2},\cdots,i_{N})}\triangleq a_{i_{\pi(1)},i_{\pi(2)},\cdots,i_ {\pi(N)}}=a_{i_{1},i_{2},\cdots,i_{N}}. \tag{2}\]  
The identity tensor of order \(N\) and dimensions \(I\) is denoted \(\mathcal{I}_{N,I}=[\delta_{i_{1},\cdots,i_{N}}]\), with \(i_{n}\in\langle I\rangle\), for \(n\in\langle N\rangle\), or simply \(\mathcal{I}\). It is a hypercubic tensor whose elements are defined using the generalized Kronecker delta  
\[\delta_{i_{1},\cdots,i_{N}}=\left\{ \begin{array}{ll}1&\text{if}\quad i_{1}= \cdots=i_{N}\\ 0&\text{otherwise}\end{array} \right..\]  
It is a diagonal tensor whose diagonal elements are equal to \(1\) and other elements to zero, which can be written as the sum of \(I\) outer products of \(N\) canonical basis vectors \(\mathbf{e}_{i}^{(I)}\) of the space \(\mathbb{R}^{I}\)  
\[\mathcal{I}_{N,I}=\sum_{i=1}^{I}\underbrace{\mathbf{e}_{i}^{(I)}\circ\cdots \circ\mathbf{e}_{i}^{(I)}}_{N\text{ terms}}.\]  
where the outer product operation is defined later in Table 9.  
A diagonal tensor \(\mathcal{X}\in\mathbb{K}^{I\times\cdots\times I}\) of order \(N\), whose diagonal elements are the entries of vector \(\mathbf{a}=[a_{1},\cdots,a_{I}]^{T}\), will be written as  
\[x_{i,i_{2},\cdots,i_{N}}=a_{i}\,\delta_{i,i_{2},\cdots,i_{N}}\,\,\Leftrightarrow \,\,\mathcal{X}=\sum_{i=1}^{I}a_{i}\underbrace{\mathbf{e}_{i}^{(I)}\circ \cdots\circ\mathbf{e}_{i}^{(I)}}_{N\text{ terms}}.\]Different matricizations, also called matrix unfoldings, can be defined for a tensor \(\mathcal{X}\in\mathbb{K}^{I_{1}\times\cdots\times I_{N}}\). Consider a partitioning of the set of modes \(\langle N\rangle\) into two disjoint ordered subsets \(\mathbb{S}_{1}\) and \(\mathbb{S}_{2}\), composed of \(p\) and \(N-p\) modes, respectively, with \(p\in\langle N-1\rangle\).