Applied Mathematical Methods

14

Singular Value Decomposition

In earlier chapters, we have studied matrices as representing linear transformations, the solution methods of systems of linear equations and the matrix eigenvalue problem. In the present chapter, we use several of the concepts developed so far, and work out a decomposition, known as the singular value decomposition, that works for an arbitrary matrix. It adds significantly to the conceptual understanding of a linear transformation on the one hand, while on the other it leads to a powerful method for diagnostics and solution of ill-posed problems.

SVD Theorem and Construction

While solving the eigenvalue problem of a matrix, we looked for a representation of the form A = UΛV⁻¹ where Λ is a diagonal matrix and ...