I want to ask the following question: Solve the following optimization problem using Lagrange multipliers for the covariance matrix $C$ and $k-1$ principal component vectors $v_1,v_2, ... v_{k-1}$.
$$ \mathbf v=\arg\max_{\mathbf v}f(\mathbf v)\,\,\text{subject to } g(\mathbf v)=\mathbf0 $$
where $f(\mathbf v)=\mathbf v^\top C\mathbf v, g_1(\mathbf v)=\|\mathbf v\|-1, g_i(\mathbf v)=\mathbf v^\top v_{i+1}$.
I know how to solve the problem without the vertical condition (i.e., we only care about $g_1$). But with the constraints of $g_2, \ldots, g_k$, I have no idea how to solve this problem.
What approach should I take to get the answer? I need a big help!