Let $f$ and $g$ be quadratic forms in real variables $x_1,\dots,x_n$, suppose that $f $ and $g$ are positive defined, is true that $f$ and $g$ are simultaneously diagonalizable?

I know that $f$ and $g$ have matrix representation $A_f$ and $A_g$ symmetric and have all eigenvalues positive, the problem then is to show that $A_f$ and $A_g$ commutes, therefore they are simultaneously diagonalizable.

So the product of symmetrical matrix is again a symmetrical matrix iff they commute, so it will be true if I can show that $A_fA_g$ is symmetric.

Am I looking in the right direction, where does the hipotesis of positive defined comes to play? There is a simplier way or a counterexample?


marked as duplicate by e.turatti, Davide Giraudo, user91500, Trevor Gunn, Harambe Jul 6 '17 at 22:57

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  • $\begingroup$ Did you look for related questions on MSE? Did you try to google your question? $\endgroup$ – Pierre-Yves Gaillard Jul 6 '17 at 15:54
  • $\begingroup$ I've found the ansewer later, the first time I couldnt find here the ansewer, but later I found. I'll close the question $\endgroup$ – e.turatti Jul 6 '17 at 16:00
  • $\begingroup$ Your "positive defined" should presumably be "positive definite". $\endgroup$ – hardmath Jul 6 '17 at 19:54