  The singular value decomposition (SVD) of a matrix is a powerful tool for
many matrix computation problems. In this paper, we consider generalizing the
standard SVD to analyze and compute the regularized solution of linear
ill-posed problems that arise from discretizing the first kind Fredholm
integral equations. For the commonly used quadrature method for discretization,
a regularizer of the form $\|x\|_{M}^2:=x^TMx$ should be exploited, where $M$
is symmetric positive definite. To handle this regularizer, we give the
weighted SVD (WSVD) of a matrix under the $M$-inner product. Several important
applications of WSVD, such as low-rank approximation and solving the least
squares problems with minimum $\|\cdot\|_M$-norm, are studied. We propose the
weighted Golub-Kahan bidiagonalization (WGKB) to compute several dominant WSVD
components and a corresponding weighted LSQR algorithm to iteratively solve the
least squares problem. All the above tools and methods are used to analyze and
solve linear ill-posed problems with the regularizer $\|x\|_{M}^2$. A
WGKB-based subspace projection regularization method is proposed to efficiently
compute a good regularized solution, which can incorporate the prior
information about $x$ encoded by the regularizer $\|x\|_{M}^2$. Several
numerical experiments are performed to illustrate the fruitfulness of our
methods.
