Applications of the singular value decomposition:
a nonnegative real number s is a singular value of matrix
m if there exist unit-length vectors u and v such
that m # v = s * u and transpose(m) # u = s * v.
u and v are then called the left-singular and
right-singular vectors for s. After svd, m, u2, s2, v2,
the elements of s2 are the singular values of m.
The pseudo-inverse of matrix m with singular value
decomposition m = u2 # mdiagonal(s2) # transpose(v2) is
equal to Mpi = v2 # mdiagonal(Spi) # transpose(u2), where
vector Spi is obtained from vector s2 by replacing each
non-zero element by its reciprocal. The solution of the linear
least-squares problem m # b = y is bs = Mpi # y, but
going through the pseudoinverse is not the most efficient method of
solving the least-squares problem.
The unit vector x that yields the smallest m # x is the
right singular vector of m corresponding to the smallest
singular value.
The null space of matrix m is spanned by the right singular
vectors corresponding to vanishing singular values of m. The
range of matrix m is spanned by the left singular vectors
corresponding to non-zero singular values of m. The rank of
matrix m equals the number of non-zero singular values.
The matrix Mr of rank r that best approximates matrix
m (by minimizing the Frobenius norm of the difference) is
Mr = u2 # mdiagonal(Sr) # transpose(v2) where Sr is
the same as s2 but has all singular values beyond the first
r set to zero.
The orthonormal matrix r closest to m (under the
Frobenius norm) is u # transpose(v) – i.e., has all singular
values replaced by 1.
The orthonormal matrix r which most closely maps a to
b (for the orthogonal Procrustes problem) is the nearest
orthonormal matrix to transpose(a) # b.