Editing Einstein notation (section)

== Common operations in this notation ==

In Einstein notation, the usual element reference <math>A_{mn}</math> for the <math>m</math>-th row and <math>n</math>-th column of matrix <math>A</math> becomes <math>{A^m}_{n}</math>. We can then write the following operations in Einstein notation as follows.

=== Inner product ===

The [[inner product]] of two vectors is the sum of the products of their corresponding components, with the indices of one vector lowered (see [[#Raising and lowering indices]]):
<math display="block">\langle\mathbf u,\mathbf v\rangle = \langle\mathbf e_i, \mathbf e_j\rangle u^i v^j = u_j v^j</math>
In the case of an [[orthonormal basis]], we have <math>u^j = u_j</math>, and the expression simplifies to:
<math display="block">\langle\mathbf u,\mathbf v\rangle = \sum_j u^j v^j = u_j v^j</math>

=== Vector cross product ===

In three dimensions, the [[cross product]] of two vectors with respect to a [[Orientation (vector space)|positively oriented]] orthonormal basis, meaning that <math>\mathbf e_1\times\mathbf e_2=\mathbf e_3</math>, can be expressed as:
<math display="block">\mathbf{u} \times \mathbf{v} = \varepsilon^i_{\,jk} u^j v^k \mathbf{e}_i</math>

Here, <math>\varepsilon^i_{\,jk} = \varepsilon_{ijk}</math> is the [[Levi-Civita symbol]]. Since the basis is orthonormal, raising the index <math>i</math> does not alter the value of <math>\varepsilon_{ijk}</math>, when treated as a tensor.

=== Matrix-vector multiplication ===

The product of a matrix {{math|''A<sub>ij</sub>''}} with a column vector {{math|''v<sub>j</sub>''}} is:
<math display="block">\mathbf{u}_{i} = (\mathbf{A} \mathbf{v})_{i} = \sum_{j=1}^N A_{ij} v_{j}</math>
equivalent to
<math display="block">u^i = {A^i}_j v^j </math>

This is a special case of matrix multiplication.

=== Matrix multiplication ===

The [[matrix multiplication|matrix product]] of two matrices {{math|''A<sub>ij</sub>''}} and {{math|''B<sub>jk</sub>''}} is:
<math display="block">\mathbf{C}_{ik} = (\mathbf{A} \mathbf{B})_{ik}  =\sum_{j=1}^N A_{ij} B_{jk}</math>

equivalent to
<math display="block">{C^i}_k = {A^i}_j {B^j}_k</math>

=== Trace ===

For a [[square matrix]] {{math|''A<sup>i</sup><sub>j</sub>''}}, the [[Trace (linear algebra)|trace]] is the sum of the diagonal elements, hence the sum over a common index {{math|''A<sup>i</sup><sub>i</sub>''}}.

=== Outer product ===

The [[outer product]] of the column vector {{math|''u<sup>i</sup>''}} by the row vector {{math|''v<sub>j</sub>''}} yields an {{math|''m''&thinsp;×&thinsp;''n''}} matrix {{math|'''A'''}}:
<math display="block">{A^i}_j = u^i v_j = {(u v)^i}_j</math>

Since {{math|''i''}} and {{math|''j''}} represent two ''different'' indices, there is no summation and the indices are not eliminated by the multiplication.

=== Raising and lowering indices ===

Given a [[tensor]], one can [[Raising and lowering indices|raise an index or lower an index]] by contracting the tensor with the [[metric tensor]], {{math|''g<sub>μν</sub>''}}. For example, taking the tensor {{math|''T<sup>α</sup><sub>β</sub>''}}, one can lower an index:
<math display="block">g_{\mu\sigma} {T^\sigma}_\beta = T_{\mu\beta}</math>

Or one can raise an index:
<math display="block">g^{\mu\sigma} {T_\sigma}^\alpha = T^{\mu\alpha}</math>