MATLAB: Are the PCA scores not othogonal after Varimax Rotation

Question

I have some data X, say 100 rows, 5 variables. I perform PCA…

[coeff,score,latent,tsquared,explained,mu] = pca(X);

I check by coeff and scores are orthogonal…

coeff'*coeff

ans =

    1.0000    0.0000    0.0000   -0.0000    0.0000
    0.0000    1.0000    0.0000   -0.0000   -0.0000
    0.0000    0.0000    1.0000   -0.0000   -0.0000
   -0.0000   -0.0000   -0.0000    1.0000   -0.0000
    0.0000   -0.0000   -0.0000   -0.0000    1.0000

corrcoef(score)

ans =

    1.0000   -0.0000    0.0000   -0.0000   -0.0000
   -0.0000    1.0000    0.0000   -0.0000    0.0000
    0.0000    0.0000    1.0000    0.0000         0
   -0.0000   -0.0000    0.0000    1.0000    0.0000
   -0.0000    0.0000         0    0.0000    1.0000

I perform what I believe to be an orthogonal Rotation:

nu_coeff=rotatefactors(coeff(:,1:3),'Method','varimax')

I check my nu coefficient matrix is othoganal:

nu_coeff'*nu_coeff

ans =

    1.0000    0.0000    0.0000
    0.0000    1.0000    0.0000
    0.0000    0.0000    1.0000

finally I test whether my new scores are orthogonal:

nu_score=X*nu_coeff

corrcoef(nu_score)

ans =

    1.0000    0.4390    0.8389
    0.4390    1.0000    0.4446
    0.8389    0.4446    1.0000

to my horror they are not….

What am I doing wrong?

Answer 1

I guess I can partially answer my own question, the short answer is scores will not be orthogonal although coefficients or "Eigen vectors" will. An easy way to see this, is as follows:

If we work in R5 space, i.e. have 5 variables and attempt the following Varimax rotation, i.e. including the complete coefficient matrix...(Eigen Vector)

nu_coeff=rotatefactors(coeff(:,1:5),'Method','varimax')

The rotated factor loadings nu_coeff will be an identify matrix

     1     0     0     0     0
     0     1     0     0     0
     0     0     1     0     0
     0     0     0     1     0
     0     0     0     0     1

Each variable loading onto one factor, the new coefficient matrix will still be orthogonal.... but each score is just the original data hence the correlation between the variables in X will be the same as the correlation between the factors in score.

corrcoef(score)==corrcoef(X)