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The rank of a tensor depends on the field over which the tensor is decomposed. It is known that some real tensors may admit a complex decomposition whose rank is strictly less than the rank of a real decomposition of the same tensor. As an example, [8] consider the following real tensor
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2.3 General rank. 3 See also. ... (1,1) tensor is a linear map. An example is the delta, ... Code of Conduct; Developers; Statistics; Cookie statement;
Thinking of matrices as tensors, the tensor rank generalizes to arbitrary tensors; for tensors of order greater than 2 (matrices are order 2 tensors), rank is very hard to compute, unlike for matrices. There is a notion of rank for smooth maps between smooth manifolds. It is equal to the linear rank of the derivative.
A tensor whose components in an orthonormal basis are given by the Levi-Civita symbol (a tensor of covariant rank n) is sometimes called a permutation tensor. Under the ordinary transformation rules for tensors the Levi-Civita symbol is unchanged under pure rotations, consistent with that it is (by definition) the same in all coordinate systems ...
It is easy to verify that = {},, …, is an orthonormal set of tensors. This means that the HOSVD can be interpreted as a way to express the tensor A {\displaystyle {\mathcal {A}}} with respect to a specifically chosen orthonormal basis B {\displaystyle B} with the coefficients given as the multidimensional array S {\displaystyle {\mathcal {S}}} .
Calculation of the invariants of rank two tensors [ edit ] In a majority of engineering applications , the principal invariants of (rank two) tensors of dimension three are sought, such as those for the right Cauchy-Green deformation tensor C {\displaystyle \mathbf {C} } which has the eigenvalues λ 1 2 {\displaystyle \lambda _{1}^{2}} , λ 2 2 ...
This mapping function projects each data pair (such as a search query and clicked web-page, for example) onto a feature space. These features are combined with the corresponding click-through data (which can act as a proxy for how relevant a page is for a specific query) and can then be used as the training data for the ranking SVM algorithm.