Editing Polar decomposition (section)

=== Derivation for normal matrices ===
If <math>A</math> is [[Normal matrix|normal]], then it is unitarily equivalent to a diagonal matrix: <math>A = V\Lambda V^*</math> for some unitary matrix <math>V</math> and some diagonal matrix <math>\Lambda ~.</math> This makes the derivation of its polar decomposition particularly straightforward, as we can then write
<math display="block">A = V\Phi_\Lambda|\Lambda|V^* = \underbrace{\left( V\Phi_\Lambda V^* \right)}_{\equiv U}\underbrace{\left(V|\Lambda|V^* \right)}_{\equiv P},</math>

where <math>|\Lambda|</math> is the matrix of absolute diagonal values, and <math>\Phi_\Lambda</math> is a diagonal matrix containing the ''phases'' of the elements of <math>\Lambda,</math> that is, <math>(\Phi_\Lambda)_{ii}\equiv \Lambda_{ii}/ |\Lambda_{ii}|</math> when <math>\Lambda_{ii} \neq 0,</math>, and <math>(\Phi_\Lambda)_{ii} = 0</math> when <math>\Lambda_{ii} = 0 ~.</math>

The polar decomposition is thus <math>A=UP,</math> with <math>U</math> and <math>P</math> diagonal in the eigenbasis of <math>A</math> and having eigenvalues equal to the phases and absolute values of those of <math>A,</math> respectively.