Consider a matrix \(X_{n\times p}\) with \(n >> p\). It induces a linear map \(X: \mathbb{R}^p \to \mathbb{R}^n\), \(v \mapsto u = Xv\). We can find a maximizer of \(f(v)=|Xv|^2\) subject to \(g(v)=|v|^2 = 1\) using Lagrange Multiplier \(\lambda\): \[X^T X v = \lambda v, \qquad v^T v = 1.\] Say \(v_1\) is a solution…