The following is an "analytic" proof of the singular value decomposition, from Tao's book "Topics in Random Matrix Theory".
I have a question on the red-marked part. I think this assertion may be related to the Lagrange multiplier method, but the fact that $u_1$ is a complex vector makes the problem complicated. How can I prove this assertion?
Basically, by identifying $U$ with $\mathbb C^n\cong\mathbb R^{2n}$, the map $u\mapsto \lVert Au\rVert^2-\sigma_1^2\lVert u\rVert^2$ is a smooth map from $\mathbb R^{2n}$ to $\mathbb R$. $u_1$ is a critical point because $\lVert Au_1\rVert^2-\sigma_1^2\lVert u_1\rVert^2=0$ is the global (hence a local) maximum. If we write in matrix form, $\lVert Au\rVert^2=u^*A^*Au$. You can check that the gradient of the map defined previously is simply $2A^*Au-2\sigma_1^2u$ after arranging the real and imaginary parts together.