# Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification
## 3 Tensors and Multilinear Forms
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Order** & **Size** & **Sets of tensors** \\ \hline \(P\) & \(\underline{I_{p}}=I_{1}\times\cdots\times I_{p}\) & \(\mathbb{K}^{I_{1}\times\cdots\times I_{p}}\triangleq\mathbb{K}^{I_{p}}\) \\ \hline \(P\) & \(\underline{I_{p}}=I_{1}\times\cdots\times I_{p}\) with \(I_{p}=I\), \(\forall p\in\langle P\rangle\) & \(\mathbb{K}^{[P;I]}\) \\ \hline \(P+N\) & \(\underline{I_{p}}\times\underline{I_{N}}=I_{1}\times\cdots\times I_{p}\times I _{1}\times\cdots\times I_{N}\) & \(\mathbb{K}^{I\times I_{N}}\) \\ \hline \multirow{3}{*}{\(P+N\)} & \(\underline{I_{p}}\times\underline{I_{N}}=I\times\cdots\times I\times J\times \cdots\times I\) & \multirow{3}{*}{\(\mathbb{K}^{[P+N;I,J]}\)} \\  & with & & \\  & \(I_{p}=I\), \(\forall p\in\langle P\rangle\) and \(J_{n}=J\), \(\forall n\in\langle N\rangle\) & \\ \hline \(2P\) & \(\underline{I_{p}}\times\underline{I_{p}}\) with \(I_{p}=I\), \(\forall p\in\langle P\rangle\) & \(\mathbb{K}^{[2P;I]}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Various sets of tensors. * The set \(\mathbb{K}^{L_{p}\times L_{p}}\) is called the set of even-order (or square) tensors of order \(2P\) and size \(L_{p}\times L_{p}\). The name of square tensor comes from the fact that the index set is divided into two identical subsets of dimension \(L_{p}\). * Analogously to matrices, tensors in the sets \(\mathbb{K}^{L_{p}\times L_{p}}\) with \(J_{p}\
eq I_{p}\) and \(\mathbb{K}^{L_{p}\times L_{N}}\) are said to be rectangular. The set \(\mathbb{K}^{L_{p}\times L_{N}}\) is called the set of rectangular tensors with index blocks of dimensions \(L_{p}\) and \(L_{N}\). The various tensor sets introduced above can be associated with scalar real-valued multilinear forms in vector variables and with homogeneous polynomials. Like in the matrix case, we will distinguish between homogeneous polynomials of degree \(P\) that depend on the components of \(P\) vector variables and those that depend on just one vector variable. A real-valued multilinear form, also called a \(P\)-linear form, is a map \(f\) such as  
\[\underset{p=1}{\overset{P}{\times}}\mathbb{R}^{I_{p}}\
i(\mathbf{x}^{(1)}, \cdots\mathbf{x}^{(P)})\ \longmapsto\ f\big{(}\mathbf{x}^{(1)},\cdots\mathbf{x}^{(P)}\big{)}\in \mathbb{R} \tag{6}\]  
that is separately linear with respect to each vector variable \(\mathbf{x}^{(p)}\) when the other variables \(\mathbf{x}^{(q)}\), for \(q\
eq p\), are fixed. Using the index convention, the multilinear form can be written for \(\mathbf{x}^{(p)}\in\mathbb{R}^{I_{p}}\), \(p\in\langle P\rangle\), as  
\[f\big{(}\mathbf{x}^{(1)},\cdots,\mathbf{x}^{(P)}\big{)}=\sum_{i_{1}=1}^{I_{1} }\cdots\sum_{i_{p}=1}^{I_{p}}a_{i_{1},\cdots,i_{p}}x_{i_{1}}^{(1)}\cdots x_{i_{ p}}^{(P)}=a_{\mathbb{I}_{p}}\prod_{p=1}^{P}x_{i_{p}}^{(p)}. \tag{7}\]  
The tensor \(\mathcal{A}\in\mathbb{R}^{I_{p}}\) is called the tensor associated with the multilinear form \(f\). Two multilinear forms are presented in Table 6, which also states the transformation corresponding to each of them, as well as the associated tensor. Table 7 recalls the definitions of bilinear/quadratic forms using the index convention, then presents the multilinear forms defined in Table 6, as well as the associated tensors from Table 5 and the corresponding homogeneous polynomials. We can make the following remarks:  
* In the same way that bilinear forms depend on two variables that do not necessarily belong to the same vector space, general real multilinear forms depend on \(P\) variables that may belong to different vector spaces: \(\mathbf{x}^{(p)}\in\mathbb{R}^{I_{p}}\).