Editing Sign function (section)

=== Polar decomposition of matrices ===
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Thanks to the [[Polar decomposition]] theorem, a matrix <math>\boldsymbol A\in\mathbb K^{n\times n}</math> (<math>n\in\mathbb N</math> and <math>\mathbb K\in\{\mathbb R,\mathbb C\}</math>) can be decomposed as a product <math>\boldsymbol Q\boldsymbol P</math> where <math>\boldsymbol Q</math> is a unitary matrix and <math>\boldsymbol P</math> is a self-adjoint, or Hermitian, positive definite matrix, both in <math>\mathbb K^{n\times n}</math>. If  <math>\boldsymbol A</math> is invertible then such a decomposition is unique and  <math>\boldsymbol Q</math> plays the role of <math>\boldsymbol A</math>'s signum. A dual construction is given by the decomposition <math>\boldsymbol A=\boldsymbol S\boldsymbol R</math> where <math>\boldsymbol R</math> is unitary, but generally different than <math>\boldsymbol Q</math>. This leads to each invertible matrix having a unique left-signum <math>\boldsymbol Q</math> and right-signum <math>\boldsymbol R</math>.

In the special case where <math>\mathbb K=\mathbb R,\ n=2,</math> and the (invertible) matrix <math>\boldsymbol A = \left[\begin{array}{rr}a&-b\\b&a\end{array}\right]</math>, which identifies with the (nonzero) complex number <math>a+\mathrm i b=c</math>, then the signum matrices satisfy <math>\boldsymbol Q=\boldsymbol P=\left[\begin{array}{rr}a&-b\\b&a\end{array}\right]/|c|</math> and identify with the complex signum of <math>c</math>, <math>\sgn c = c/|c|</math>.  In this sense, polar decomposition generalizes to matrices the signum-modulus decomposition of complex numbers.