Given the following constrained optimization problem
$$ \underset{W_1, W_2}{\text{argmax}} \quad W_1^T M W_2 \quad\text{subject to}\quad W_1^T W_1 = I, \quad W_2^TW_2 = I $$
The closed form solution $(W_1^*, W_2^*)$ is found through SVD decomposition of matrix $M$
$$M = W_1^* D (W_2^*)^T$$
I applied KKT conditions but could not derive the final solution. How could we prove this result?
UPDATES:
Lagrangian equation:
$$L(W_1, W_2) = W_1^TMW_2 - \lambda_1 (W_1^TW_1 - 1) - \lambda_2 (W_2^TW_2 - 1)$$
Using KKT conditions, I arrived at the following equations:
$$MW_2 = 2\lambda_1 W_1$$
$$W_1^TM = 2\lambda_2 W_2^T$$