0

I'm trying to solve a problem and need to prove that product of two positive definite matrices is diagonalizable.

I know matrices are diagonalizable when they are symmetric. How do I prove that product of 2 positive definite matrices is symmetric?

thanks!

luxerhia
  • 3,538
MSquare
  • 43

1 Answers1

3

If $A,B$ are positive definite and $A^{1/2}$ denotes the (unique) positive definite square root of $A$, then $AB$ is similar to $A^{1/2}BA^{1/2}$, which is symmetric. Because $A^{1/2}BA^{1/2}$ is diagonalizable, $AB$ must be diagonalizable as well.

Ben Grossmann
  • 225,327
  • Oh this is clever. I like it. It took me a second to realize that similar is the key here, not necessarily unitarily equivalent (and I suspect they aren't unless $A$ and $B$ commute). – Cameron Williams May 14 '21 at 13:58
  • 1
    @CameronWilliams Of course they can't be unitarily equivalent in that case: anything orthogonally/unitarily similar to a symmetric/Hermitian matrix must be symmetric/Hermitian itself, and for this problem $AB$ is symmetric if and only if $A,B$ commute. – Ben Grossmann May 14 '21 at 14:01
  • Does this proof apply to general positive definite A, B not being symmetric as well because it is similar instead of equivalent? – MSquare May 14 '21 at 14:10
  • 1
    @MSquare It does not. Non-symmetric positive definite matrices $A$ will not necessarily have a positive definite square root, and the fact that $A^{1/2}BA^{1/2}$ is symmetric relies on the symmetry of $B$. – Ben Grossmann May 14 '21 at 14:11
  • Is there any way to prove this for general positive definite A and B as well? – MSquare May 14 '21 at 14:33
  • @MSquare I doubt that it is true without the assumption that $A,B$ are symmetric. If there is a way, then I don't know about it. – Ben Grossmann May 14 '21 at 14:34
  • got it! thank you! – MSquare May 14 '21 at 14:34