Editing Einstein notation (section)

=== Abstract description ===

The virtue of Einstein notation is that it represents the [[Invariant (mathematics)|invariant]] quantities with a simple notation.

In physics, a [[Scalar (physics)|scalar]] is invariant under transformations of basis. In particular, a [[Lorentz scalar]] is invariant under a [[Lorentz transformation]]. The individual terms in the sum are not. When the basis is changed, the ''components'' of a vector change by a [[linear transformation]] described by a matrix. This led Einstein to propose the convention that repeated indices imply the summation is to be done.

As for covectors, they change by the [[inverse matrix]]. This is designed to guarantee that the linear function associated with the covector, the sum above, is the same no matter what the basis is.

The value of the Einstein convention is that it applies to other [[vector space]]s built from {{math|''V''}} using the [[tensor product]] and [[dual space|duality]]. For example, {{math|''V'' ⊗&thinsp;''V''}}, the tensor product of {{math|''V''}} with itself, has a basis consisting of tensors of the form {{math|1='''e'''<sub>''ij''</sub> = '''e'''<sub>''i''</sub> ⊗ '''e'''<sub>''j''</sub>}}. Any tensor {{math|'''T'''}} in {{math|''V'' ⊗&thinsp;''V''}} can be written as:
<math display="block">\mathbf{T} = T^{ij}\mathbf{e}_{ij}.</math>

{{math|''V''&hairsp;*}}, the dual of {{math|''V''}}, has a basis {{math|'''e'''<sup>1</sup>}}, {{math|'''e'''<sup>2</sup>}}, ..., {{math|'''e'''<sup>''n''</sup>}} which obeys the rule
<math display="block">\mathbf{e}^i (\mathbf{e}_j) = \delta^i_j.</math>
where {{math|''δ''}} is the [[Kronecker delta]]. As
<math display="block">\operatorname{Hom}(V, W) = V^* \otimes W</math>
the row/column coordinates on a matrix correspond to the upper/lower indices on the tensor product.