pseudoinverse

For rectangular matrices, you need to be careful as to whether you are taking a left pseudoinverse or a right pseudoinverse as the following is also a right pseudoinverse:

More on Bürger et al 2006 « Climate Audit 2006

Suppose that R is a rectangular matrix and S is a square matrix, if you do a right Moore-Penrose pseudoinverse, but the following does NOT hold

More on Bürger et al 2006 « Climate Audit 2006

The Moore-Penrose pseudoinverse is familiar to people who use linear regressions:

More on Bürger et al 2006 « Climate Audit 2006

At this point, we can expand the Moore-Penrose right pseudoinverse:

More on Bürger et al 2006 « Climate Audit 2006

Bürger et al used Moore-Penrose pseudoinverse inverse notation, which I had not used.

More on Bürger et al 2006 « Climate Audit 2006

I got my solutions using pseudoinverse + iterations, under-determined set of linear equations + constraint on the solution, I’m quite sure there is a better way ..

Inversions from Partial Correlation Coefficients « Climate Audit 2007