# Principal Component analysis by eigenvalue decomposition.

I do know how to perform PCA by using SVD but I am unaware about how to use eigenvalue decomposition of X(transpose)*X matrix. I found a paper online which explains the approach to perform PCA by using eigen decomposition. But, it does not clearly explain all the stuff. Is there a simple explanation for this. Thanks.

• As far as I know, SVD is usually defined (and certainly first taught) in terms of eigen decomposition. Do you define the SVD in terms of eigenvalues and eigenvectors of $X^TX$? If not, then how do you define it? – Omnomnomnom Apr 20 '15 at 12:06

## 1 Answer

Doing SVD on the matrix $X$ is (almost) the same as doing the eigenvalue-decomposition of the covariance matrix $X^T X$ (Note that $X$ is symmetric and hence diagonalizable using eigenvalue decomposition). In general, doing SVD on $X$ is better than forming the covariance matrix and then doing eigenvalue decomposition, which could lead to additional loss of accuracy.