Editing Trace (linear algebra) (section)

===Traces of special kinds of matrices===
{{bulleted list
| The trace of the {{math|''n'' × ''n''}} [[identity matrix]] is the dimension of the space, namely {{mvar|n}}.
<math display="block">\operatorname{tr}\left(\mathbf{I}_n\right) = n</math>
This leads to [[Dimension (vector space)#Trace|generalizations of dimension using trace]].
| The trace of a [[Hermitian matrix]] is real, because the elements on the diagonal are real.
| The trace of a [[permutation matrix]] is the number of [[Fixed point (mathematics)|fixed points]] of the corresponding permutation, because the diagonal term {{math|''a''<sub>''ii''</sub>}} is 1 if the {{math|''i''}}th point is fixed and 0 otherwise.
| The trace of a [[Projection_(linear_algebra)|projection matrix]] is the dimension of the target space.
<math display="block">\begin{align}
                                  \mathbf{P}_\mathbf{X} &= \mathbf{X}\left(\mathbf{X}^\mathsf{T} \mathbf{X}\right)^{-1} \mathbf{X}^\mathsf{T} \\[3pt]
\Longrightarrow
    \operatorname{tr}\left(\mathbf{P}_\mathbf{X}\right) &= \operatorname{rank}(\mathbf{X}).
\end{align}</math>
The matrix {{math|'''P<sub>X</sub>'''}} is idempotent.
| More generally, the trace of any [[idempotent matrix]], i.e. one with {{math|1='''A'''<sup>2</sup> = '''A'''}}, equals its own [[rank (linear algebra)|rank]].
| The trace of a [[nilpotent matrix]] is zero.
{{pb}}
When the characteristic of the base field is zero, the converse also holds: if {{math|1=tr('''A'''<sup>''k''</sup>) = 0}} for all {{mvar|k}}, then {{math|'''A'''}} is nilpotent.
{{pb}}
When the characteristic {{math|''n'' > 0}} is positive, the identity in {{mvar|n}} dimensions is a counterexample, as <math>\operatorname{tr}\left(\mathbf{I}_n^k\right) = \operatorname{tr}\left(\mathbf{I}_n\right) = n \equiv 0</math>, but the identity is not nilpotent.
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