Editing Trace (linear algebra) (section)

=== Bilinear forms ===
The [[bilinear form]] (where {{math|'''X'''}}, {{math|'''Y'''}} are square matrices)
<math display="block">B(\mathbf{X}, \mathbf{Y}) = \operatorname{tr}(\operatorname{ad}(\mathbf{X})\operatorname{ad}(\mathbf{Y}))</math>
: where <math>\operatorname{ad}(\mathbf{X})\mathbf{Y} = [\mathbf{X}, \mathbf{Y}] = \mathbf{X}\mathbf{Y} - \mathbf{Y}\mathbf{X}</math>
: and for orientation, if <math>\operatorname{det} \mathbf{Y} \ne 0 </math>
:: then <math>\operatorname{ad}(\mathbf{X}) = \mathbf{X} - \mathbf{Y}\mathbf{X}\mathbf{Y}^{-1} ~.</math>

<math> B(\mathbf{X}, \mathbf{Y})</math> is called the [[Killing form]]; it is used to classify [[Lie algebra]]s.

The trace defines a bilinear form:
<math display="block">(\mathbf{X}, \mathbf{Y}) \mapsto \operatorname{tr}(\mathbf{X}\mathbf{Y}) ~.</math>

The form is symmetric, non-degenerate<ref group=note>This follows from the fact that {{math|1=tr('''A'''*'''A''') = 0}} [[if and only if]] {{math|1='''A''' = '''0'''}}.</ref> and associative in the sense that:
<math display="block">\operatorname{tr}(\mathbf{X}[\mathbf{Y}, \mathbf{Z}]) = \operatorname{tr}([\mathbf{X}, \mathbf{Y}]\mathbf{Z}).</math>

For a complex simple Lie algebra (such as {{math|<math>\mathfrak{sl}</math><sub>''n''</sub>}}), every such bilinear form is proportional to each other; in particular, to the Killing form{{Citation needed|reason=Either a source or proof is needed|date=June 2022}}.

Two matrices {{math|'''X'''}} and {{math|'''Y'''}} are said to be ''trace orthogonal'' if
<math display="block">\operatorname{tr}(\mathbf{X}\mathbf{Y}) = 0.</math>

There is a generalization to a general representation <math>(\rho,\mathfrak{g},V)</math> of a Lie algebra <math>\mathfrak{g}</math>, such that <math>\rho</math> is a homomorphism of Lie algebras <math>\rho: \mathfrak{g} \rightarrow \text{End}(V).</math> The trace form <math>\text{tr}_V</math> on <math>\text{End}(V)</math> is defined as above. The bilinear form
<math display="block">\phi(\mathbf{X},\mathbf{Y}) = \text{tr}_V(\rho(\mathbf{X})\rho(\mathbf{Y}))</math>
is symmetric and invariant due to cyclicity.