# PCA, relation between the error and variance

As is known, the rank-1 PCA aims to solve the following optimization problem $$\min_{x\in\mathbb{R}^d}\quad -x^T \Sigma x\quad\quad\quad \text{s.t.}\quad \Vert x\Vert_{2}=1,$$ where $$\Sigma\in\mathbb{S}^{d}$$ is the covariance matrix. Thus the optimum $$x^*$$ of the PCA problem is the top unit eigenvector of $$\Sigma$$. Given an approximation $$\tilde{x}$$ (normalized), the error between the $$\tilde{x}$$ and $$x^*$$ is measured by the sine function $$\sin^{2}(\tilde{x}, x^*) = 1-(\tilde{x}^T x^*)^2.$$ I was wondering does there exist any analytical relationship between the objective function $$\tilde{x}^\top\Sigma \tilde{x}$$ and the error $$\sin^2(\tilde{x}, x^*)$$? Any help appreciated.

• Thx! This answer really helps! Can this result be extended to rank-k PCA? In this case, the sine of the angle might be written as $\Vert sin(X, X^*)\Vert_{F}^2 = k-\Vert X^T X^*\Vert_{F}^2$. May 14 at 16:04
• I have post the rank-$k$ case as another new question: link thx for help! May 15 at 8:02