Everything about Rank Of A Tensor totally explained
In
mathematics, the
rank of a
tensor T is a term suffering under
overloading. In this case
rank may mean, frequently, either of two things.
It
may mean the total number of indices required to write down the components of
T (which is the sum of the number of covariant and of the number of contravariant indices). Expressed by means of the
tensor product of
multilinear algebra, this is one more than the number of tensor product signs required to express
T.
Or, in the case where the rank in the preceding sense is 2, in other words a
dyadic tensor, it may mean the same as in
linear algebra: that is, the
rank of a matrix. This meaning is possibly the intended one, whenever the array of components is two-dimensional.
For example, a
dyadic product has rank 2 in the first sense, and rank 1 or 0 in the second sense.
It is now preferred, in order to avoid ambiguities, to use the terminology of "tensor order" to denote the number of indices, and "tensor rank" to designate the number of simple tensors necessary to decompose a tensor. Hence the definition of rank is consistent with Linear Algebra.
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