In deriving the eigenvectors for PCA, the vector is subject to the condition that it should be of unit length. Why is this so?

**Answer**

The main aim of Principal Component Analysis (PCA) is to look for the directions on Rp that maximize the variance of the projected random vector X=(X1,…,Xp). Specifically, the first PC can be defined as the unit vector v(1)∈Rp such that

v(1)=argmax

If you allow vectors that are not of unit norm in the maximization problem, then you will not get a proper solution, since variance of the projection can become arbitrarily large as long as the norm of the vector increases. For example, if w=\lambda v, with v,w\in\mathbb{R}^p and \lambda\to\infty, then

\mathbb{V}\mathrm{ar}\big[w^TX\big]=\lambda^2\mathbb{V}\mathrm{ar}\big[v^TX\big]\to\infty\quad (\text{if }\mathbb{V}\mathrm{ar}\big[v^TX\big]\neq0).

This is the reason why you need an standardization of unit norm to constraint the search and avoid improper solutions.

**Attribution***Source : Link , Question Author : Suhas Lohit , Answer Author : epsilone*