Coming from singular value decomposition. There exists a simpler decomposition that the singular value decomposition of any square matrix: polar decomposition. It consists of
$$ A=WP $$where $W$ 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 singular 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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Author of the notes: Antonio J. Pan-Collantes
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