  This paper provides an advanced mathematical theory of the Generalized
Singular Value Decomposition (GSVD) and its applications. We explore the
geometry of the GSVD which provides a long sought for ellipse picture which
includes a horizontal and a vertical multiaxis. We further propose that the
GSVD provides natural coordinates for the Grassmann manifold. This paper proves
a theorem showing how the finite generalized singular values do or do not
relate to the singular values of $AB^\dagger$.
  We then turn to the applications arguing that this geometrical theory is
natural for understanding existing applications and recognizing opportunities
for new applications. In particular the generalized singular vectors play a
direct and as natural a mathematical role for certain applications as the
singular vectors do for the SVD. In the same way that experts on the SVD often
prefer not to cast SVD problems as eigenproblems, we propose that the GSVD,
often cast as a generalized eigenproblem, is rather best cast in its natural
setting.
  We illustrate this theoretical approach and the natural multiaxes (with
labels from technical domains) in the context of applications where the GSVD
arises: Tikhonov regularization (unregularized vs regularization), Genome
Reconstruction (humans vs yeast), Signal Processing (signal vs noise), and
statistical analysis such as ANOVA and discriminant analysis (between clusters
vs within clusters.) With the aid of our ellipse figure, we encourage in the
future the labelling of the natural multiaxes in any GSVD problem.
