Let $T_1, \ldots, T_n$ by real symmetric positive definite matrices, with eigenvalues bounded below by $\mu > 0$. Can I say $$ \frac{x^T T_1 T_2 \ldots T_n x}{x^T x} \geq \mu^n $$ I know you can say this for a product of two matrices (e.g., Eigenvalues of the product of two symmetric matrices), but what about the case where I have $n \geq 2$ matrices in the product? If these matrices commute the result is straightforward, but I'm interested in the case where these matrices don't necessarily commute.

Let $T_1, \ldots, T_n$ by real symmetric positive definite matrices, with eigenvalues bounded below by $\mu > 0$. Can I say $$ \frac{x^T T_1 T_2 \ldots T_n x}{x^T x} \geq \mu^n $$ If these matrices commute the result is straightforward, but I'm interested in the case where these matrices don't necessarily commute.

Edit: Not sure that you can say this for the $n=2$ case either.