Prove that maximum value is the largest eigenvalue

Question

Given a symmetric positive definite matrix $A \in \mathbb{R}^{n \times n}$ and a full rank matrix $B \in \mathbb{R}^{n \times n}$. Prove that the maximum value the following optimization problem is the largest eigenvalue of $B^{-1}B^{-T}A$.

$$\begin{array}{ll} \text{maximize} & x^T A \, x\\ \text{subject to} & \|Bx\| = 1 \end{array}$$

Can you help me with this one?

Answer 1

Let $y=Bx$. Then $x=B^{-1}y$ and the problem becomes: Prove that the maximum value of $y^T(B^{-T}AB^{-1})y$ subject to $||y||=1$ is the largest eigenvalue of $B^{-T}AB^{-1}$. It is not hard to attain the conclusion.