  With the advent of very large redshift surveys of tens to hundreds of
thousands of galaxies reliable techniques for automatically determining galaxy
redshifts are becoming increasingly important. The most common technique
currently in common use is the cross-correlation of a galactic spectrum with a
set of templates. This series of papers presents a new method based on
Principal Component Analysis. The method generalizes the cross-correlation
approach by replacing the individual templates by a simultaneous linear
combination of orthogonal templates. This effectively eliminates the mismatch
between templates and data and provides for the possibility of better error
estimates. In this paper, the first of a series, the basic mathematics are
presented along with a simple demonstration of the application.
