# Principal axes, what are they and how to decide them? [duplicate]

I am reading a book about data mining and am currently in a chapter about Principal Component Analysis. But I am not sure from the explanation in the book what the principal axes are and how to find them.

If I have the feature vector $X = \{(-3,-1,-1),(0,-1,0),(-2,-1,2),(1,-1,3)\}$, how do I find the principal axes?

Thanks!

With assume that you know the exactly reason for using PCA ,First of all assume that we have a our X matrix contains our data ( we have d features):
$$X = \begin{array}{cc} x_1^1 & x_2^1 & ... & x_d^1 \\ x_2^1 & x_2^2 & ... & x_d^2 \\ ...\\ x_n^1 & x_n^2 & ... & x_d^n\\ \end{array}$$
$$E[X] = \mu = [\mu_1,...,\mu_d]$$ and compute matrix of all feature Covariance : $$Cov(x) = E[(x-\mu)(x-\mu)^T]$$ now we have Covariance matrix we should compute eigenvectors and eigenvalues of Covariance Matrix , for this d*d matrix we have d eigenvalues , sort them from highest to lowest , now here is the PCA trick , we can ignore some of eigenvalues that are below a threshold because the eigenvectors belongs to them represents the feature that have lowest variance between data , and that feature is not very handy for classify objects :