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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?