Let A and B be commutative symmetric positive definite matrices. The goal of the question was to show that $||x||_{A,B}=\sqrt{<Ax,Bx>}$ is a norm in $\mathbf{R}^n$, where $<x,y>$ is the canonical scalar product.
For that, my idea was to show that $(x,y)_{A,B}=<Ax,By>$ is a scalar product, and then using Cauchy-Schwarz inequality I can show that $||x||_{A,B}=\sqrt{<Ax,Bx>}$ is a norm (since the only real problem is proving triangle inequality).
When trying to show that $(x,y)_{A,B}$ is a scalar product, I managed to show that it's bilinear and symmetric, but I can't show that $<Ax,Bx> > 0$ for all $x \in \mathbf{R}^n$.
Since $<Ax,Bx>=<BAx,x>$, all that is left to show is:
"Let A and B be commutative symmetric positive definite matrices, then AB is positive definite"
Furthermore, if there's another easier way to show that $||x||_{A,B}$ is a norm, please show me.
$A$ and $B$ commute by assumption and are diagonalisable because they are symmetric positive definite, so they are simultaneously diagonalisable. This implies every eigenvalue of $AB$ is the product of an eigenvalue of $A$ and an eigenvalue of $B$ and so is positive.