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The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as a matrix, [3] and can be determined from Gaussian elimination for instance. The rank of an order 3 or higher tensor is however often very difficult to determine, and low rank decompositions of tensors are sometimes of great practical interest. [4]
On the other hand, a randomly sampled complex tensor of the same size will be a rank-1 tensor with probability zero, a rank-2 tensor with probability one, and a rank-3 tensor with probability zero. It is even known that the generic rank-3 real tensor in R 2 ⊗ R 2 ⊗ R 2 {\displaystyle \mathbb {R} ^{2}\otimes \mathbb {R} ^{2}\otimes \mathbb ...
The total number of indices is also called the order, degree or rank of a tensor, [2] [3] [4] although the term "rank" generally has another meaning in the context of matrices and tensors. Just as the components of a vector change when we change the basis of the vector space, the components of a tensor also change under such a transformation.
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 ...
The rank of a tensor is the minimum number of rank-one tensor that must be summed to obtain the tensor. A rank-one tensor may be defined as expressible as the outer product of the number of nonzero vectors needed to obtain the correct order. Dyadic tensor A dyadic tensor is a tensor of order two, and may be represented as a square matrix. In ...
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
More precisely, matrices are tensors of type (1,1), having one row index and one column index, also called covariant order 1 and contravariant order 1; see Tensor (intrinsic definition) for details. The tensor rank of a matrix can also mean the minimum number of simple tensors necessary to express the matrix as a linear combination, and that ...
Hooke's law has a symmetric fourth-order stiffness tensor with 81 components (3×3×3×3), but because the application of such a rank-4 tensor to a symmetric rank-2 tensor must yield another symmetric rank-2 tensor, not all of the 81 elements are independent. Voigt notation enables such a rank-4 tensor to be represented by a 6×6 matrix ...