# Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification
## 3 Tensors and Multilinear Forms
A general matrix unfolding formula was given by [71] as follows  
\[\mathbf{X}_{\mathbb{S}_{1}\mathbb{S}_{2}}=\sum_{i_{1}=1}^{I_{1}}\cdots\sum_{i_ {N}=1}^{I_{N}}x_{i_{1},\cdots,i_{N}}\bigg{(}\underset{n\in\mathbb{S}_{1}}{ \otimes}\mathbf{e}_{i_{n}}^{(I_{n})}\bigg{)}\bigg{(}\underset{n\in\mathbb{S}_{2 }}{\otimes}\mathbf{e}_{i_{n}}^{(I_{n})}\bigg{)}^{T}\in\mathbb{K}^{I_{1}\times \mathbb{I}_{2}}, \tag{3}\]  
where \(\mathbf{e}_{i_{n}}^{(I_{n})}\) is the \(i_{n}\)-th vector of the canonical basis of \(\mathbb{R}^{I_{n}}\), and \(J_{n_{1}}=\prod\limits_{n\in\mathbb{S}_{n_{1}}}I_{n}\), for \(n_{1}=1\) and \(2\). We say that \(\mathbf{X}_{\mathbb{S}_{1}\mathbb{S}_{2}}\) is a matrix unfolding of \(\mathcal{X}\) along the modes of \(\mathbb{S}_{1}\) for the rows and along the modes of \(\mathbb{S}_{2}\) for the columns, with \(\mathbb{S}_{1}\cap\mathbb{S}_{2}=\mathcal{O}\) and \(\mathbb{S}_{1}\cup\mathbb{S}_{2}=\langle N\rangle\). For instance, in the case of a third-order tensor \(\mathcal{X}\in\mathbb{K}^{I\times I\times K}\), we have six flat unfoldings and six tall unfoldings. For \(\mathbb{S}_{1}=1\) and \(\mathbb{S}_{2}=\{2,3\}\), we have the following mode-1 flat unfolding \(\mathbf{X}_{I\times JK}\triangleq\mathbf{X}_{\mathbb{I};\{2,3\}}\), while for \(\mathbb{S}_{1}=\{2,3\}\) and \(\mathbb{S}_{2}=1\) we obtain the following mode-1 tall unfolding \(\mathbf{X}_{J\times J\downarrow}\triangleq\mathbf{X}_{\{2,3\};\}=\mathbf{X}_{I \times JK}^{T}\). Vectorized forms of \(\mathcal{X}\in\mathbb{K}^{I_{1}\times\cdots\times I_{N}}\) are obtained by combining the modes in a given order. Thus, a lexicographical vectorization gives the vector \(\mathbf{y}\triangleq\mathbf{x}_{I_{1}\cdots I_{N}}\) with element \(x_{i_{1},\cdots,i_{N}}\) at the position \(m=\overline{i_{1}i_{2}\cdots i_{N}}\) in \(\mathbf{y}\), i.e., \(y_{m}=x_{i_{1},\cdots,i_{N}}\triangleq x_{\mathbb{I}_{N}}\), with [72]  
\[\overline{i_{1}i_{2}\cdots i_{N}}\triangleq i_{N}+\sum_{n=1}^{N-1}(i_{n}-1) \prod_{k=n+1}^{N}I_{k}. \tag{4}\]  
By convention, the order of the dimensions in a product \(\prod_{n=1}^{N}I_{n}\triangleq I_{1}\cdots I_{N}\) associated with the index combination \(\overline{i_{1}i_{2}\cdots i_{N}}\) follows the order of variation of the indices \((i_{1},\cdots,i_{N})\), with \(i_{1}\) varying more slowly than \(i_{2}\), which in turn varies more slowly than \(i_{3}\), etc. The Frobenius norm of \(\mathcal{X}\in\mathbb{K}^{I_{1}\times\cdots\times I_{N}}\) is the square root of the inner product of the tensor with itself, i.e.,  
\[\|\mathcal{X}\|_{F}=\sqrt{\langle\mathcal{X},\mathcal{X}\rangle}=\Bigg{(} \sum_{i_{1}=1}^{I_{1}}\cdots\sum_{i_{N}=1}^{I_{N}}|x_{i_{1},\cdots,i_{N}}|^{2} \Bigg{)}^{1/2}. \tag{5}\]  
Table 5 presents various sets of tensors that will be considered in this paper, with the notation introduced in [70]. We can make the following remarks about the sets of tensors defined in Table 5:  
* For \(P=N=1\), the set \(\mathbb{K}^{[2;I,J]}\) is the set \(\mathbb{K}^{I\times J}\) of (real or complex) matrices of size \(I\times J\). * The set \(\mathbb{K}^{[P;I]}\) is also denoted \(\mathbb{K}^{I^{p}}\) or \(\mathbb{T}^{p}(K^{I})\) by some authors.