TABLE OF CONTENTS Before we turn to the numerical regularization methods in the following chapters, it is convenient to summarize the "canonical decompositions" which are important theoretical as well as computational tools in connection with rank-deficient and discrete ill-posed problems. Along with this discussion, it is natural to briefly summarize methods for transforming regularization problems in general form into standard form. We also describe how the singular value expansion (SVE) of a kernel can be computed by means of the singular value decomposition (SVD).
The superior numerical "tools" for analysis of rank-deficient and discrete ill-posed problems are the (ordinary) SVD of A and its generalization to two matrices, the generalized SVD (GSVD) of the matrix pair ( A, L); see [154, 2.5.3 4 and 8.7.3]. The SVD reveals all the difficulties associated with the ill-conditioning of the matrix A, while the GSVD of ( A, L) yields important insight into regularization problems involving both the matrix A and the regularization matrix L, such as in (1.14).
The use of the SVD and the GSVD in the analysis of discrete ill-posed problems goes back to Hanson [201] and Varah [352], [353]. The SVD has many similarities with the SVE discussed in 1.2.1, and the early history of both is described in [324].
Let A 
be a rectangular or square matrix, and assume for ease of presentation that m ? n. Then the SVD of A
