Suppose there are two matrices $\textit{A}, \textit{B}$. Note that $\textit{A}$ is positive-semi definite, $\textit{B}$ is positive definite matrix, both are of size $N\times N$ and symmetric matrix.
If I solve the problem $Ax = \lambda Bx$, then there is a property which is eigendecomposition such that
$$ X^TAX = \Lambda\\ X^TBX = I_N, $$ where, $\Lambda = \text{diag}(\lambda_1,\cdots,\lambda_N)$ is eigenvalue matrix, the values are on only diagonal elements with descending order, $\lambda_1 \geq \cdots \geq \lambda_N$, $X$ is eigenvector matrix with the order corresponding $\Lambda$. $I_N$ is $N\times N$ identity matrix.
I would like to know the following derivation is correct.
Assume that I solve the problem
$$ \alpha Ax = \lambda \beta Bx, $$ where, $\alpha$, $\beta$ are constants except $0$.
I end up with this relationship,
$$ \Big(\sqrt{\frac{\alpha}{\beta}}X\Big)^TA\sqrt{\frac{\alpha}{\beta}}X = {\frac{\alpha}{\beta}}\Lambda\\ \Big(\sqrt{\frac{\alpha}{\beta}}X\Big)^T B\sqrt{\frac{\alpha}{\beta}}X = I_N. $$
Can this be correct? If there needs to put all derivations here, I would like to.