Editing Normal matrix (section)

== Equivalent definitions ==
It is possible to give a fairly long list of equivalent definitions of a normal matrix. Let {{mvar|A}} be a {{math|''n'' × ''n''}} complex matrix. Then the following are equivalent:
# {{mvar|A}} is normal.
# {{mvar|A}} is [[diagonalizable matrix|diagonalizable]] by a unitary matrix. 
# There exists a set of eigenvectors of {{mvar|A}} which forms an orthonormal basis for {{math|'''C'''<sup>''n''</sup>}}. 
# <math>\left\| A \mathbf{x} \right\| = \left\| A^* \mathbf{x} \right\|</math> for every {{Math|'''x'''}}.
# The [[Frobenius norm]] of {{mvar|A}} can be computed by the eigenvalues of {{mvar|A}}: <math display="inline"> \operatorname{tr} \left(A^* A\right) = \sum_j \left| \lambda_j \right|^2 </math>.
# The [[Hermitian matrix|Hermitian]] part {{math|{{sfrac|1|2}}(''A'' + ''A''<sup>*</sup>)}} and [[Skew-Hermitian matrix|skew-Hermitian]] part {{math|{{sfrac|1|2}}(''A'' − ''A''<sup>*</sup>)}} of {{mvar|A}} commute.
# {{math|''A''<sup>*</sup>}} is a polynomial (of degree {{math|≤ ''n'' − 1}}) in {{mvar|A}}.<ref group="lower-alpha">Proof: When <math>A</math> is normal, use [[Lagrange polynomial|Lagrange's interpolation]] formula to construct a polynomial <math>P</math> such that <math>\overline{\lambda_j} = P(\lambda_j)</math>, where <math>\lambda_j</math> are the eigenvalues of <math>A</math>.</ref>
# {{math|1=''A''<sup>*</sup> = ''AU''}} for some unitary matrix {{mvar|U}}.<ref>{{Harvp|Horn|Johnson|1985|page=109}}</ref>
# {{mvar|U}} and {{mvar|P}} commute, where we have the [[polar decomposition]] {{math|1=''A'' = ''UP''}} with a unitary matrix {{mvar|U}} and some [[positive-definite matrix|positive semidefinite matrix]] {{mvar|P}}.
# {{mvar|A}} commutes with some normal matrix {{mvar|N}} with distinct{{clarify|date=September 2023}} eigenvalues.
# {{math|1=''σ<sub>i</sub>'' = {{abs|''λ<sub>i</sub>''}}}} for all {{math|1 ≤ ''i'' ≤ ''n''}} where {{mvar|A}} has [[singular values]] {{math|''σ''<sub>1</sub> ≥ ⋯ ≥ ''σ<sub>n</sub>''}} and has eigenvalues that are indexed with ordering {{math|{{abs|''λ''<sub>1</sub>}} ≥ ⋯ ≥ {{abs|''λ<sub>n</sub>''}}}}.<ref>{{Harvp|Horn|Johnson|1991|page=[https://archive.org/details/topicsinmatrixan0000horn/page/157 157]}}</ref>

Some but not all of the above generalize to normal operators on infinite-dimensional Hilbert spaces. For example, a bounded operator satisfying (9) is only [[quasinormal operator|quasinormal]].