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Let $S_n^{++}(\Bbb R)$ be the set of positive definite symmetric matrices with real coefficients, $A_n(\Bbb R)$ the set of skew-symmetric matrices with real coefficients. Show that if $A\in S_n^{++}(\Bbb R)$ and $B \in A_n(\Bbb R)$ therefore $AB$ is diagonalizable in $\Bbb M_n(\Bbb C)$.

If $AB=BA$ I know how to get the result. But I'm stuck if $AB \neq BA$... Does someone have a hint?

Michelle
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1 Answers1

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Hint: prove that $AB$ is similar to a skew-symmetric matrix with real coefficients.

user1551
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  • Thanks a lot! May I ask how you came up with this idea? Do you have any references that may help me see this as something "natural"? – Michelle Jan 06 '21 at 15:35
  • @Michelle It's gut feeling, but it is also natural to consider what diagonalisable matrices $AB$ is similar to. – user1551 Jan 06 '21 at 16:00
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    @Michelle Another bit of intuition: whenever a positive definite matrix is involved, it is helpful to ask "is there anything that I can do with $A^{1/2}$?" – Ben Grossmann Jan 06 '21 at 16:05