Editing Matrix decomposition (section)

=== Polar decomposition ===
{{main|Polar decomposition}}
*Applicable to: any square complex matrix ''A''.
*Decomposition: <math>A=UP</math> (right polar decomposition) or <math>A=P'U</math> (left polar decomposition), where ''U'' is a [[unitary matrix]] and ''P'' and ''P''' are [[positive semidefinite matrix|positive semidefinite]] [[Hermitian matrices]].
*Uniqueness: <math>P</math> is always unique and equal to <math>\sqrt{A^*A}</math> (which is always hermitian and positive semidefinite). If <math>A</math> is invertible, then <math>U</math> is unique.
*Comment: Since any Hermitian matrix admits a spectral decomposition with a unitary matrix, <math>P</math> can be written as <math>P=VDV^*</math>. Since <math>P</math> is positive semidefinite, all elements in <math>D</math> are non-negative. Since the product of two unitary matrices is unitary, taking <math>W=UV</math>one can write <math>A=U(VDV^*)=WDV^* </math> which is the singular value decomposition. Hence, the existence of the polar decomposition is equivalent to the existence of the singular value decomposition.