One of the most useful constructs in numerical analysis is the singular value decomposition (SVD) of a matrix, which uses orthogonal transformations to diagonalize the matrix. It has long been known how to extend the SVD to the singular value expansion (SVE) of a compact operator.

In this talk, I will explain why the SVD and SVE are so useful when analyzing linear inverse problems and show how to extend the concept of the SVE to general bounded linear operators defined on Hilbert space.

In certain problems involving a pair of operators, the SVE is not enough. We need a similar construct that allows us to represent the two operators in a consistent form---essentially, to simultaneously diagonalize the two operators. If time permits, I will present the generalized singular value expansion and briefly show how it simplifies the analysis of certain inverse problems.