Many implementations of PCA use singular value decomposition to calculate eigenvectors and eigenvalues. SVD is given by the following equation:
Columns of U are called left singular vectors of the data matrix, the columns of V are its right singular vectors, and the diagonal entries of 
are its singular values. Left singular vectors are the eigenvectors of the covariance matrix and the diagonal element of 
are the square roots of the eigenvalues of the covariance matrix.
Before proceeding with SVD, it would be advisable to understand a few advantages and important points about SVD:
- SVD can be applied even on rectangular matrices; whereas, eigenvalues are defined only for square matrices. The equivalent of eigenvalues obtained through the SVD method are called singular values, and vectors obtained equivalent to eigenvectors are known as singular vectors. However, as they are rectangular in nature, we need to have left singular vectors and right singular vectors respectively for their dimensions.
- If a matrix A has a matrix of eigenvectors P that is not invertible, then A does not have an eigen decomposition. However, if A is m x n real matrix with m > n, then A can be written using a singular value decomposition.
- Both U and V are orthogonal matrices, which means UT U = I (I with m x m dimension) or VT V = I (here I with n x n dimension), where two identity matrices may have different dimensions.
is a non-negative diagonal matrix with m x n dimensions.
Then computation of singular values and singular vectors is done with the following set of equations:
In the first stage, singular values/eigenvalues are calculated with the equation. Once we obtain the singular/eigenvalues, we will substitute to determine the V or right singular/eigen vectors:
Once we obtain the right singular vectors and diagonal values, we will substitute to obtain the left singular vectors U using the equation mentioned as follows:
In this way, we will calculate the singular value decompositions of the original system of equations matrix.