ABSTRACT
Exercise 5-20: Write the singular value decomposition of Mt explicitly for the case m<n. Then find the corresponding decomposition of M by transposing it.
We can use this theorem to show that any matrix equation can be reduced to an equation that contains a diagonal matrix; this often simplifies the analysis of such equations. Consider the matrix equation
with M again an n×m matrix with n>m; x, a vector of length m, and y, a vector of length n. Then, using the singular value decomposition,
If none of the singular values of D is zero, then D is nonsingular and
Therefore, using the orthogonality properties of U and V,
This provides a simple method for finding the inverse of M. The computational problem, of course, is to find the singular value decomposition of the given matrix M. Press et al show how to do this computationally.
