Detailed balance

2014-01-21T19:10:19Z

129.219.43.68: /* Reversible Markov chains */

{{Unreferenced|date=November 2009}}
A '''two-vector''' is a [[tensor]] of type (2,0) and it is the [[dual space|dual]] of a [[two-form]], meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars).

The [[tensor product]] of a pair of [[Vector (geometric)|vector]]s is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of vectors, especially a linear combination of tensor products of pairs of basis vectors. If '''f''' is a two-vector, then
:<math> \mathbf{f} = f^{\alpha \beta} \, \vec e_\alpha \otimes \vec e_\beta </math>
where the ''f <sup>α β</sup>'' are the components of the two-vector. Notice that both indices of the components are [[covariance and contravariance|contravariant]]. This is always the case for two-vectors, by definition.

An example of a two-vector is the inverse ''g<sup>μ ν</sup>'' of the [[metric tensor]].

The components of a two-vector may be represented in a matrix-like array. However, a two-vector, as a tensor, should not be confused with a [[Matrix (mathematics)|matrix]], since a matrix is a linear function
:<math> M : V \rightarrow V </math>
which [[Map (mathematics)|maps]] vectors to vectors, whereas a two-vector is a linear functional
:<math> \mathbf{f} : \tilde{V} \rightarrow V </math>
which maps [[one-form]]s to vectors. In this sense, a matrix, considered as a tensor, is a [[mixed tensor]] of type (1,1) even though of the same rank as a two-vector.

{{DEFAULTSORT:Two-Vector}}
[[Category:Tensors]]

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Detailed balance