Tensors

What are tensors ?

Vectors are generalizations of scalars in higher dimension. Matrices are generalizations of vectors in another mode. Tensors can be seen as generalizations of matrices in higher modes.

Multlinear tensors

$$T\equiv [T_{i_1,...,i_d}]=\sum _{i_1,\dots ,i_n}{T}_{i_1,\dots ,i_d}{x}_{1,i_1}\otimes \cdots \otimes x_{d,i_d}\in E_1\otimes \cdots \otimes E_d$$

Symmetric tensors

$$F=\sum _{|\alpha |=d}{F}_{\alpha }{x}_1^{{\alpha }_1}\cdots x_n^{{\alpha }_n}=\sum _{|\alpha |=d}{F}_{\alpha }{\mathbf{x}}^{\alpha }\in S^d(E)$$

Skew-symmetric tensors

$$\Psi =\sum _{i_1<\cdots <i_d}{\Psi }_{i_1,\dots ,i_d}{x}_{i_1}\wedge \cdots \wedge x_{i_d}\in {\wedge }^d(E)$$

Multi-symmetric tensors

$$F=\sum _{|{\alpha }_1|=d_1,\dots ,|{\alpha }_m|=d_m}{F}_{{\alpha }_1,\dots ,{\alpha }_m}{\mathbf{x}}_1^{{\alpha }_1}\cdots {\mathbf{x}}_m^{{\alpha }_m}\in S^{d_1}(E_1)\otimes \cdots \otimes S^{d_m}(E_m)$$