Group Theory and Linear Algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 23 September 2014

Lecture 19: Polar decomposition

Let f:VV be a linear transformation. Let ,:V×V be a positive definite Hermitian form. Show that the following are equivalent.

(a) f is self adjoint and all eigenvalues are positive.
(b) There exists g:VV such that g is self adjoint and f=g2.
(c) There exists h:VV such that f=hh*.
(d) f is self adjoint and f(v),v0 for all vV.

Proof.

Let AGLn(). Then there exist P, diagonalisable with positive eigenvalues, and U, unitary, such that A=PU.

Idea of proof.

Notes and References

These are a typed copy of Lecture 19 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on September 6, 2011.

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