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Linear Algebra, Optimisation and Numerical Methods / The Full Singular Value Decomposition

The Polar Decomposition

This closing note of Topic 6 assembles two classes of operator built earlier in the unit — isometries and positive semi-definite operators — into a single factorisation of any operator: A=WPA=WP, an isometry times a positive semi-definite operator. This is the matrix analogue of writing a complex number z=eiθrz=e^{i\theta}r (a pure rotation times a non-negative magnitude), and it falls directly out of the SVD with almost no extra work.
Learning objectives. By the end of this note you should be able to:
  • state the polar decomposition and construct it from the SVD;
  • explain the complex-number analogy precisely;
  • verify the polar decomposition on a small example;
  • explain why P=A⊤AP=\sqrt{A^\top A} is the "stretch" part and WW is the "rotation" part.

In this lesson

  1. The decomposition

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