A very powerful set of techniques dealing with sets of equations or matrices that are either singular or numerically very close to singular is the socalled singular value decomposition (SVD). SVD allows one to diagnose the problems in a given matrix and provides numerical answer as well. Any m x n matrix a (m >= n) can be written as the product of an m x n columnorthogonal matrix u, an n x n diagonal matrix with positive or zero elements, and the transpose of an n x n orthogonal matrix v:
where
and
The diagonal elements of matrix w are the singular values of matrix a and nonnegative numbers. If m = n, matrices u and v are square and the inverse matrix of a becomes
where
since
As shown in Figure 6, matrices u and v are orthogonal and their inverse matrices are equal to their transposes. One can have a problem in [4] and [5] if one or more w's are zero or very close to zero. Matrix a is singular in this case. Solution for a System of Linear Equations For a system of linear equations,
where a = a square matrix, and x & y = column matrices, [4] can be used to obtain its solution:
where
The diagonal elements of matrix w' are given as
where e = the singularity threshold. In other words, if w_{i} is zero or close to zero (smaller than e), one must replace 1/w_{i} (infinity) with 0 in [9]. e depends on the precision of the hardware. When matrix a is singular, [7] will not have a solution, but replacing 1/w_{i} with 0 will provide the closest x that minimizes
in the least square sense. Interestingly, [8], [9] & [10] can be also used in an overdetermined system where the number of equations exceed that of the parameters. Reference & Related Literature Press, W.M., Flannery, B.P., Teukolsky, S.A., & Vetterling, W.T. (1986). Numerical recipes: The art of scientific computing. New York, NY: Cambridge University Press. Lay, D.C. (1996). Linear algebra and its applications, 2nd ed. Reading, MA: AddisonWesley.

© YoungHoo Kwon, 1998 