SVD: Singular Value Decomposition of a Matrix

In matlib: Matrix Functions for Teaching and Learning Linear Algebra and Multivariate Statistics

Singular Value Decomposition of a Matrix

Description

Compute the singular-value decomposition of a matrix X either by Jacobi rotations (the default) or from the eigenstructure of X'X using Eigen. Both methods are iterative. The result consists of two orthonormal matrices, U, and V and the vector d of singular values, such that X = U diag(d) V'.

Usage

SVD(
  X,
  method = c("Jacobi", "eigen"),
  tol = sqrt(.Machine$double.eps),
  max.iter = 100
)

Arguments

X	a square symmetric matrix
method	either "Jacobi" (the default) or "eigen"
tol	zero and convergence tolerance
max.iter	maximum number of iterations

Details

The default method is more numerically stable, but the eigenstructure method is much simpler. Singular values of zero are not retained in the solution.

Value

a list of three elements: d– singular values, U– left singular vectors, V– right singular vectors

Author(s)

John Fox and Georges Monette

See Also

svd, the standard svd function

Eigen

Examples

C <- matrix(c(1,2,3,2,5,6,3,6,10), 3, 3) # nonsingular, symmetric
C
SVD(C)

# least squares by the SVD
data("workers")
X <- cbind(1, as.matrix(workers[, c("Experience", "Skill")]))
head(X)
y <- workers$Income
head(y)
(svd <- SVD(X))
VdU <- svd$V %*% diag(1/svd$d) %*%t(svd$U)
(b <- VdU %*% y)
coef(lm(Income ~ Experience + Skill, data=workers))

matlib documentation built on Oct. 3, 2024, 1:09 a.m.