Matrix polar decomposition

In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to the polar form of a nonzero complex number z as ${\displaystyle z=re^{i\theta }\,}$ where r is the absolute value of z (a positive real number), and ${\displaystyle e^{i\theta }}$ is an element of the circle group.

The polar decomposition of a square complex matrix A is a matrix decomposition of the form

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