Editing Trace (linear algebra) (section)

== Generalizations ==
The concept of trace of a matrix is generalized to the [[trace class]] of [[compact operator]]s on [[Hilbert space]]s, and the analog of the [[Frobenius norm]] is called the [[Hilbert–Schmidt operator|Hilbert–Schmidt]] norm.

If <math>K</math> is a trace-class operator, then for any [[orthonormal basis]] <math>\{e_n\}_{n=1}</math>, the trace is given by
<math display="block">\operatorname{tr}(K) = \sum_n \left\langle e_n, Ke_n \right\rangle,</math>
and is finite and independent of the orthonormal basis.<ref>{{cite book | first=G. | last=Teschl | title=Mathematical Methods in Quantum Mechanics | series=Graduate Studies in Mathematics | volume=157 | date=30 October 2014 | publisher=American Mathematical Society | isbn=978-1470417048 | edition=2nd}}</ref>

The [[partial trace]] is another generalization of the trace that is operator-valued. The trace of a linear operator <math>Z</math> which lives on a product space <math>A\otimes B</math> is equal to the partial traces over <math>A</math> and <math>B</math>:
<math display="block">\operatorname{tr}(Z) = \operatorname{tr}_A \left(\operatorname{tr}_B(Z)\right) = \operatorname{tr}_B \left(\operatorname{tr}_A(Z)\right).</math>

For more properties and a generalization of the partial trace, see [[Traced monoidal category|traced monoidal categories]].

If <math>A</math> is a general [[associative algebra]] over a field <math>k</math>, then a trace on <math>A</math> is often defined to be any [[linear functional|functional]] <math>\operatorname{tr}:A\to k</math> which vanishes on commutators; <math>\operatorname{tr}([a,b])=0</math> for all <math>a,b\in A</math>. Such a trace is not uniquely defined; it can always at least be modified by multiplication by a nonzero scalar.

A [[supertrace]] is the generalization of a trace to the setting of [[superalgebra]]s.

The operation of [[tensor contraction]] generalizes the trace to arbitrary tensors.

Gomme and Klein (2011) define a matrix trace operator <math>\operatorname{trm}</math> that operates on [[block matrix|block matrices]] and use it to compute second-order perturbation solutions to dynamic economic models without the need for [[tensor notation]].<ref>{{cite journal |author=P. Gomme, P. Klein |title=Second-order approximation of dynamic models without the use of tensors |journal=Journal of Economic Dynamics & Control |volume=35 |year=2011 |issue=4 |pages=604–615 |doi=10.1016/j.jedc.2010.10.006 }}</ref>