Rank of a tensor

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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.