  The generalized singular value decomposition (GSVD) of a matrix pair $\{A,
L\}$ with $A\in\mathbb{R}^{m\times n}$ and $L\in\mathbb{R}^{p\times n}$
generalizes the singular value decomposition (SVD) of a single matrix. In this
paper, we provide a new understanding of GSVD from the viewpoint of SVD, based
on which we propose a new iterative method for computing nontrivial GSVD
components of a large-scale matrix pair. By introducing two linear operators
$\mathcal{A}$ and $\mathcal{L}$ induced by $\{A, L\}$ between two
finite-dimensional Hilbert spaces and applying the theory of singular value
expansion (SVE) for linear compact operators, we show that the GSVD of $\{A,
L\}$ is nothing but the SVEs of $\mathcal{A}$ and $\mathcal{L}$. This result
characterizes completely the structure of GSVD for any matrix pair with the
same number of columns. As a direct application of this result, we generalize
the standard Golub-Kahan bidiagonalization (GKB) that is a basic routine for
large-scale SVD computation such that the resulting generalized GKB (gGKB)
process can be used to approximate nontrivial extreme GSVD components of $\{A,
L\}$, which is named the gGKB\_GSVD algorithm. We use the GSVD of $\{A, L\}$ to
study several basic properties of gGKB and also provide preliminary results
about convergence and accuracy of gGKB\_GSVD for GSVD computation. Numerical
experiments are presented to demonstrate the effectiveness of this method.
