Questions in the process of finding the first principal component vector.

Question

The process of finding the first principal component, $\mathbf{w}_{(1)}$, is explained in PCA. However, I do not understand why the following two equations are equivalent. $$ \underset{\lVert\mathbf{w}\rVert=1}{\operatorname{arg\,max}}\left\{\mathbf{w}^T\mathbf{X}^T\mathbf{X}\mathbf{w}\right\} = \operatorname{arg\,max}\left\{\frac{\mathbf{w}^T\mathbf{X}^T\mathbf{X}\mathbf{w}}{\mathbf{w}^T\mathbf{w}}\right\} $$ The site says it is because $\mathbf{w}$ is a unit vector. Is there a rigorous proof?

Answer 1

The norm of $\mathbf{w}$ can be written as $\sqrt{\mathbf{w}^T\mathbf{w}}$. So, the expression $\frac{\mathbf{w}^T\mathbf{X}^T\mathbf{X}\mathbf{w}}{\mathbf{w}^T\mathbf{w}}$ can be rewritten as $\frac{\mathbf{w}^T\mathbf{X}^T\mathbf{X}\mathbf{w}}{||\mathbf{w}||^2}=\frac{\mathbf{w}^T}{||\mathbf{w}||}X^TX\frac{\mathbf{w}}{||\mathbf{w}||}$ which is of the form on the left hand side.