simple tensor

The tensor productPlanetmathPlanetmathPlanetmath ( \PMlinkescapephrasetensor product UโŠ—V of two vector spacesMathworldPlanetmath U and V is another vector space which is characterised by being universalPlanetmathPlanetmathPlanetmath for bilinear maps on Uร—V. As part of this package, there is an operationMathworldPlanetmath โŠ— on vectors such that ๐ฎโŠ—๐ฏโˆˆUโŠ—V for all ๐ฎโˆˆU and ๐ฏโˆˆV, and the primary subject of this article is the image of that operation.

Definition 1.

The element ๐ฐโˆˆUโŠ—V is said to be a simple tensor if there exist ๐ฎโˆˆU and ๐ฏโˆˆV such that ๐ฐ=๐ฎโŠ—๐ฏ.

More generally, the element ๐ฐโˆˆW=U1โŠ—โ‹ฏโŠ—Uk is said to be a simple tensor (with respect to the decomposition U1โŠ—โ‹ฏโŠ—Uk of W) if there exist ๐ฎiโˆˆUi for i=1,โ€ฆ,k such that ๐ฐ=๐ฎ1โŠ—โ‹ฏโŠ—๐ฎk.

For this definition to be interesting, there must also be tensors which are not simple, and indeed most tensors arenโ€™t. In order to illustrate why, it is convenient to consider the tensor product of two finite-dimensional vector spaces U=๐’ฆm and V=๐’ฆn over some field ๐’ฆ. In this case one can let UโŠ—V=๐’ฆmร—n (the vector space of mร—n matrices), since ๐’ฆmร—n is isomorphic to any generic construction of UโŠ—V and the tensor product of two spaces is anyway only defined up to isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath. Furthermore considering elements of U and V as column vectorsMathworldPlanetmath, the tensor product of vectors can be defined through


where โ‹… denotes the productPlanetmathPlanetmathPlanetmath of two matrices (in this case an mร—1 matrix by a 1ร—n matrix). As a very concrete example of this,


One reason the simple tensors in UโŠ—V cannot exhaust this space (provded m,nโฉพ2) is that there are essentially only m+n-1 degrees of freedom in the choice of a simple tensor, but mโขn dimensions ( in the space UโŠ—V as a whole. Hence

๐’ฆmโŠ—๐’ฆnโ‰ {๐ฎโŠ—๐ฏย ๐ฎโˆˆ๐’ฆm,๐ฏโˆˆ๐’ฆn}โ€ƒโ€ƒwhenย m,nโฉพ2.

How can one to understand the non-simple tensors, then? In general, they are finite sums of simple tensors. One way to see this is from the theorem that UโŠ—V has a basis consisting of products of pairs of basis vectors.

Theorem 2 (tensor product basis (

Let U and V be vector spaces over K with bases {ei}iโˆˆI and {fj}jโˆˆJ respectively. Then {eiโŠ—fj}(i,j)โˆˆIร—J is a basis for UโŠ—V.

Expressing some arbitrary ๐ฐโˆˆUโŠ—V as a linear combinationMathworldPlanetmath


with respect to such a basis immediately produces the decomposition


as a sum of simple tensors, but this decomposition is often far from optimally short. Let ๐ž1=(10)โˆˆ๐’ฆ2 and ๐ž2=(01)โˆˆ๐’ฆ2. The tensor ๐ž1โŠ—๐ž1+๐ž2โŠ—๐ž2=(1001) is not simple, but as it happens the tensor ๐ž1โŠ—๐ž1+๐ž1โŠ—๐ž2+๐ž2โŠ—๐ž1+๐ž2โŠ—๐ž2=(1111)=(11)โŠ—(11) is simple. In general it is not trivial to find the simplest way of expressing a tensor as a sum of simple tensors, so there is a name for the length of the shortest such sum.

Definition 3.

The rank of a tensor ๐ฐ is the smallest natural numberMathworldPlanetmath n such that ๐ฐ=๐ฐ1+โ‹ฏ+๐ฐn for some set of n simple tensors ๐ฐ1, โ€ฆ, ๐ฐn.

In particular, the zero tensor has rank 0, and all other simple tensors have rank 1.

  • Warning.ย ย There is an entirely different concept which is also called โ€˜the rank of a tensor (โ€™, namely the number of componentsPlanetmathPlanetmath (factors) in the tensor product forming the space in which the tensor lives. This latter โ€˜rankโ€™ concept does not generalise โ€˜rank of a matrix (โ€™. The โ€˜rankโ€™ of Definitionย 3 does generalise โ€˜rank of a matrixโ€™. (It also generalises rank of a quadratic form (


one way

One area where the distinction between simple and non-simple tensors is particularly important is in Quantum Mechanics, because the state space of a pair of quantum systems is in general the tensor product of the state spaces of the component systems. When the combined state is a simple tensor ๐ฐ=๐ฎโŠ—๐ฏ, then that state can be understood as though one subsystem has state ๐ฎ and the other state ๐ฏ, but when the combined state ๐ฐ is a non-simple tensor ๐ฎ1โŠ—๐ฏ1+๐ฎ2โŠ—๐ฏ2 then the full system cannot be understood by considering the two subsystems in isolation, even if there is no apparent interaction between them. This situation is often described by saying that the two subsystems are entangled, or using phrases such as โ€œeither U is in state ๐ฎ1 and V is in state ๐ฏ1, or else U is in state ๐ฎ2 and V is in state ๐ฏ2.โ€ Entanglement is an important part of that which makes quantum systems different from probabilistic classical systems. The physical interpretationsMathworldPlanetmathPlanetmath are often mind-boggling, but the mathematical meaning is no more mysterious than โ€˜non-simple tensorโ€™.

Entanglement can also be a useful concept for understanding pure mathematics. One reason that the comultiplication ฮ”:CโŸถCโŠ—C of a coalgebra C cannot simply be replaced in the definition by two maps ฮ”L,ฮ”R:CโŸถC that compute the โ€˜leftโ€™ and โ€˜rightโ€™ parts of ฮ” is that value of ฮ” may be entangled, in which case one left part ฮ”Lโข(c) and one right part ฮ”Rโข(c) cannot fully encode ฮ”โข(c).

Title simple tensor
Canonical name SimpleTensor
Date of creation 2013-03-22 15:26:07
Last modified on 2013-03-22 15:26:07
Owner lars_h (9802)
Last modified by lars_h (9802)
Numerical id 5
Author lars_h (9802)
Entry type Definition
Classification msc 15A69
Synonym tensor rank
Synonym entangled
Related topic TensorProduct
Related topic BasicTensor
Defines rank