Singular value decomposition. Polar decomposition

I don't know how I have not learned this until today, because it looks a very basic fact. Given any real square matrix, \(A\), we can write
\[
A=U \Sigma V
\]
where \(U\) and \(V\) are orthogonal matrices and \(\Sigma\) is a diagonal matrix with non negative entries.

The meaning of this is easy: any linear transformation of a vector space can be obtained by a rigid transformation (rotation or reflection), followed by a scale change (different scales could be applied in every axis) and finally followed by another rigid transformation.





Moreover, there exists a simpler decomposition of any square matrix: polar decomposition. It consists of
\[
A=WP
\]
where \(U\) is an orthogonal matrix (rotation or reflection) and \(P\) is a positive definite symmetric matrix. That is, every linear transformation of a vector space can be decomposed as a scale change, not necessarily in the main axis direction, and not necessarily with equal scales, and a rigid transformation.

We can obtain it from the single value decomposition:
\[
A=U\Sigma V=UVV^{\perp} \Sigma V
\]
and we take \(W=UV\) and \(P=V^{\perp} \Sigma V\)

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