I am unable to think of a way to prove this result: Show that every square matrix is a product of a hermitian matrix and an unitary matrix.

I want to prove it using the spectral theorem.

Thanks for the help.


This is usually done via the "polar decomposition" of a matrix; more information on that is given here. This answer details what that looks like for real matrices; a similar proof applies in the complex case, where the $T$'s are replaced by $*$'s.

  • $\begingroup$ How can the spectral theorem be used to prove it? $\endgroup$ – Tejas P Sep 1 '17 at 14:05
  • $\begingroup$ The spectral theorem provides a convenient way to show that $A^*A$ has a unique positive definite square root. $\endgroup$ – Omnomnomnom Sep 1 '17 at 14:17
  • $\begingroup$ Could you please sketch the entire proof? $\endgroup$ – Tejas P Sep 1 '17 at 14:49
  • $\begingroup$ @TejasP I have no intention of doing so. This is a standard result which you can find in most linear algebra textbooks. $\endgroup$ – Omnomnomnom Sep 1 '17 at 14:51

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