# Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification
## 2 Notation and Index Convention  
Scalars, column vectors, matrices, and tensors are denoted by lower-case, boldface lower-case, boldface upper-case, and calligraphic letters, e.g., \(x\), \(\mathbf{x}\), \(\mathbf{x}\), \(\mathcal{X}\), respectively. We denote by \(a_{i,r}\) the \((i,r)\) element and by \(\mathbf{A}_{r}\) (resp. \(\mathbf{A}_{i. }\)) the \(r\)th column (resp. \(i\)th row) of \(\mathbf{A}\in\mathbb{C}^{l\times R}\). \(\mathbf{I}_{R}\) denotes the identity matrix of size \(R\times R\). The transpose, complex conjugate, transconjugate, and Moore-Penrose pseudo-inverse operators are represented by \((. )^{T}\), \((-)^{*}\), \((\cdot)^{H}\) and \((\cdot)^{\dagger}\), respectively. The operator \(\text{diag}(\cdot)\) forms a diagonal matrix from its vector argument, while \(D_{i}(\mathbf{A})\) stands for a diagonal matrix holding the \(i\)th row of \(\mathbf{A}\in\mathbb{C}^{I\times R}\) on the diagonal. The operator \(\mathcal{T}_{M+N-1,\,N}(\cdot)\) forms a \((M+N-1)\times N\) Toeplitz matrix from its vector argument \(\mathbf{x}\in\mathbb{C}^{M}\), whose first column and row are, respectively, \(\left( \begin{array}{cc}x_{1}&\cdots&x_{M}&\mathbf{0}_{N-1}^{T}\end{array}  \right)^{T}\) and \(\left( \begin{array}{cc}x_{1}&\mathbf{0}_{N-1}^{T}\end{array} \right)\). Given \(\mathbf{Y}\in\mathbb{C}^{I\times I}\), the vec and unvec operators are defined such that: \(\mathbf{y}=\text{vec}(\mathbf{Y})\in\mathbb{C}^{II}\leftrightarrow\mathbf{Y}= \text{unvec}(\mathbf{y})\in\mathbb{C}^{I\times J}\), where the order of dimensions in the product \(JI\) is linked to the order of variation of the indices, with the column index \(j\) varying more slowly than the row index \(i\). The outer, Kronecker and Khatri-Rao products are denoted by \(\circ\), \(\otimes\) and \(\diamond\), respectively. Table 3 summarizes the notation used for sets of indices and dimensions [70]. We now introduce the index convention which allows eliminating the summation symbols in formulae involving multi-index variables. For example, \(\sum_{i=1}^{I}a_{i}b_{i}\) is simply written as \(a_{i}b_{i}\). Note there are two differences relative to [PERSON]'s summation convention:  
* Each index can be repeated more than twice in an expression;
* Ordered index sets are allowed. The index convention can be interpreted in terms of two types of summation, the first associated with the row indices (superscripts) and the second associated with the column indices (subscripts), with the following rules [70]:  
* The order of the column indices is independent of the order of the row indices;
* Consecutive row and column indices (or index sets) can be permuted. In Table 4, we give some examples of vector and matrix products using index convention, where \(\mathbf{e}_{ij}\triangleq\mathbf{e}_{i}^{(I)}\otimes\mathbf{e}_{j}^{(I)}, \mathbf{e}_{i}^{j}\triangleq\mathbf{e}_{i}^{(I)}\otimes(\mathbf{e}_{j}^{(I)} )^{T},\mathbf{e}_{ik}^{j}\triangleq\mathbf{e}_{i}^{(I)}\otimes\mathbf{e}_{k }^{(K)}\otimes(\mathbf{e}_{j}^{(I)})^{T}\). \begin{table}
\begin{tabular}{c} \hline \hline \(\mathbf{u}\in\mathbb{K}^{I},\mathbf{v}\in\mathbb{K}^{J},\mathbf{w}\in\mathbb{ K}^{K}\) \\ \hline \(\mathbf{u}\otimes\mathbf{v}=u_{i}v_{i}\mathbf{e}_{ij}\in\mathbb{K}^{IJ}\) \\ \hline \(\mathbf{u}\otimes\mathbf{v}^{T}=u_{i}v_{i}\mathbf{e}_{i}^{j}\in\mathbb{K}^{I \times J}\) \\ \hline \(\mathbf{u}\otimes\mathbf{v}^{T}\otimes\mathbf{w}=u_{i}v_{i}v_{i}\mathbf{e}_{ ik}^{j}\in\mathbb{K}^{IK\times J}\) \\ \hline \(\mathbf{A}\in\mathbb{K}^{I\times J},\mathbf{B}\in\mathbb{K}^{J\times K}, \mathbf{C}\in\mathbb{K}^{K\times J}\) \\ \hline \(\mathbf{AB}=\sum_{i=1}^{I}\sum_{k=1}^{K}(\sum_{j=1}^{J}a_{ij}b_{jk})\mathbf{e}_ {i}^{k}=a_{ij}b_{jk}\mathbf{e}_{i}^{k}\in\mathbb{K}^{I\times K}\) \\ \hline \(\mathbf{AC}^{T}=a_{ij}c_{kj}\mathbf{e}_{i}^{k}\in\mathbb{K}^{I\times K}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Vector and matrix products using the index convention.