linear algebra - Is it possible to get the transformation PrincipalComponents uses to transform data?

As far as I can see PrincipalComponents[data] just gives data in the principal component basis. The problem is that after this, I would like to transform new data into the same basis, which requires that I know the actual principal components.

There is the option of calculating Eigenvectors[Covariance[data]] (given the data is centered on the origin) but for the eigenvectors I get this way the signs are usually different from the ones that the PrincipalComponents[] function uses.

Is there an easy way of getting the actual transformation used or transforming new data into the PC basis of the previous dataset?

Answer

You can do it with FindGeometricTransform[]:

data = {{13.2, 200, 58, 21.2}, {10, 263, 48, 44.5}, {8.1, 294, 80, 31}, 
        {8.8, 190, 50, 19.5}, {9, 276, 91, 40.6}, {7.9, 204, 78, 38.7}, 
        {3.3, 110, 77, 11.1}, {5.9, 238, 72, 15.8}, {15.4, 335, 80, 31.9}, {17.4, 211, 60, 25.8}};
{error, f} = FindGeometricTransform[PrincipalComponents[data], data];
Norm[PrincipalComponents[data][[1]] - f[data[[1]]]]
(*
  0.8
*)