# Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification
## 4 Tensor Operations and Decompositions
### Multiplications with Tensors  
In Table 8, we present three types of multiplication with tensors, using the notation of Table 3 and the index convention: mode-\(p\), mode-\((p,n)\), and Einstein products. The multiplication \(\times_{p}\), called mode-\(p\) or Tucker product, corresponds to a summation over the index \(i_{p}\) associated with the mode \(p\) of the \(P\)th-order tensor \(\mathcal{X}\) and the second index of \(\mathbf{A}\), giving a tensor of order \(P-1\), and size \(I_{1}\times\cdots\times I_{p-1}\times I_{p+1}\times\cdots\times I_{P}\). The mode-\((p,n)\) product, denoted \(\times_{p}^{n}\), corresponds to a contraction operation performed for two arbitrary modes \((p,n)\), such as: \(I_{p}=J_{n}=K\). This multiplication gives a tensor of order \(P+N-2\) and size \(I_{1}\times\cdots\times I_{p-1}\times I_{p+1}\times\cdots\times I_{P}\times J _{1}\times\cdots\times J_{n-1}\times J_{n+1}\times\cdots\times J_{N}\). The [PERSON] product, denoted \(\mathcal{A}\star_{N}\mathcal{X}\), of the tensors \(\mathcal{A}\in\mathbb{K}^{I_{p}\times I_{N}}\) of order \(P+N\) and \(\mathcal{X}\in\mathbb{K}\mathbb{l}_{N^{\times}\mathbb{k}_{Q}}\) of order \(N+Q\) corresponds to a contraction along the \(N\) shared indices \(\mathbf{\hat{l}}_{N}\), associated with the \(N\) last modes of \(\mathcal{A}\) and the \(N\) first modes of \(\mathcal{X}\). The tensor \(\mathcal{A}\) can be interpreted as a multilinear operator associated with a multilinear transformation applied to the tensor \(\mathcal{X}\). The Einstein product will be used in section 6 for defining multilinear systems. Table 9 presents a few examples of outer products of vectors, matrices, and tensors, indicating the order and the space to which the tensors resulting from the products belong. \begin{table}
\begin{tabular}{c c c} \hline
**Tensors** & **Operations** & **Definitions** \\ \hline \(\mathcal{X}\in\mathbb{K}^{I_{p}},\mathcal{A}\in\mathbb{K}^{I_{p}}\) & \(\mathcal{Y}=\mathcal{X}\times_{p}\mathbf{A}\) & \(y_{1_{p}\cdots j_{p-1}\times i_{p+1}\cdots j_{p}}=\sum_{p}a_{ijr}x_{\mathbf{i}p}= a_{ijr}x_{\mathbf{i}p}\) \\ \hline \(\mathcal{X}\in\mathbb{K}^{I_{p}},\mathbf{u}\in\mathbb{K}^{p}\) & \(\mathcal{Y}=\mathcal{X}\times_{p}\mathbf{u}^{T}\) & \(y_{1_{p}\cdots j_{p-1}\times i_{p+1}\cdots j_{p-1}}=\sum_{p}a_{ijr}x_{\mathbf{i} p}=y_{r}x_{\mathbf{i}p}\) \\ \hline \(\mathcal{X}\in\mathbb{K}^{I_{p}},\mathcal{Y}\in\mathbb{K}^{I_{N}}\) & \(Z=\mathcal{X}\times_{p}^{n}Y\) & \(z_{1_{p}\cdots j_{p-1}\times i_{p+1}\cdots j_{p-1}\times i_{p-1}\cdots i_{p+ 1}\cdots i_{p-N}}=\) \\ with \(I_{p}=I_{n}=K\) & & \(\sum_{k_{p}\cdots i_{p-1}\times i_{p-1}\times i_{p-1}\cdots i_{p-1}\times i_{p- 1}\cdots i_{p-1}\times i_{p-1}\cdots i_{p-N}}\) \\ \hline \(\mathcal{A}\in\mathbb{K}^{I_{p}\times I_{N}},\mathcal{X}\in\mathbb{K}^{I_{N} =\mathbb{k}_{Q}}\) & \(\mathcal{Y}=\mathcal{A}\star_{N}\mathcal{X}\) & \(y_{\mathbf{i}p\mathbf{i}\mathbf{k}_{Q}}=\sum_{\mathbf{i}p\mathbf{i}\mathbf{k}_{ Q}}=\sum_{\mathbf{i}p\mathbf{i}\mathbf{k}_{Q}}=\sum_{\mathbf{i}p\mathbf{i} \mathbf{k}_{Q}}=\sum_{\mathbf{i}p\mathbf{i}\mathbf{k}_{Q}}=\sum_{\mathbf{i}p \mathbf{i}\mathbf{k}_{Q}}\) \\ \hline \end{tabular}
\end{table} 
Table 8: Different types of multiplication with tensors.