How do i obtain only the first principal component?

For certain measurements i need to obtain only the numeric value of the first principal component from the matrix. Can someone please tell me how do i go about it?

[sidra]

Any suggestions?

Answer 1

[Jan]

I'm not sure, if I fully understand your question. I doubt however, that there is a straightforward method for calculating the eigenvector corresponding to the largest eigenvalue of the covariance matrix without calculating all eigenvalues first (at least not for non-sparse matrices).

If you want the first principal component of the (m x n)-matrix A containing m measurements as row vectors you would in general do the following:

A = randn(100, 20);    % artificial sample matrix
c_A = cov(A);
[V, ~], eigs( c_A ); 
p_1 = V( :, 1 );

which gives you the direction of the first principal component in the variable p_1.

[sidra]

Thank you so much jan :)

Answer 2

[Andrew Knyazev]

https://www.mathworks.com/matlabcentral/fileexchange/48-lobpcg-m can be used as the method for calculating the eigenvector corresponding to the largest eigenvalue of the covariance matrix without calculating all eigenvalues, or even without explicitly calculating the covariance matrix itself.