Sign function
Polar decomposition of matrices

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

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