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In special relativity, one of the simplest non-trivial examples of a four-tensor is the four-displacement = (,,,) = (,,,) a four-tensor with contravariant rank 1 and covariant rank 0. Four-tensors of this kind are usually known as four-vectors.
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 ...
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 tensors are classified according to their type (n, m), where n is the number of contravariant indices, m is the number of covariant indices, and n + m gives the total order of the tensor. For example, a bilinear form is the same thing as a (0, 2)-tensor; an inner product is an example of a (0, 2)-tensor, but not all (0, 2)-tensors are inner ...
Matrix rank should not be confused with tensor order, which is called tensor rank. Tensor order is the number of indices required to write a tensor , and thus matrices all have tensor order 2. 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 ...
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]
It is closely related to the Ricci tensor. Being a second rank tensor in four dimensions, the energy–momentum tensor may be viewed as a 4 by 4 matrix. The various admissible matrix types, called Jordan forms cannot all occur, as the energy conditions that the energy–momentum tensor is forced to satisfy rule out certain forms.
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 ...