matlib (version 0.9.1)

SVD: Singular Value Decomposition of a Matrix

Description

Compute the singular-value decomposition of a matrix $X$ from the eigenstructure of $X^{\prime }X$. The result consists of two orthonormal matrices, $U$, and $V$ and the vector $d$ of singular values, such that $X=Udiag(d)V^{\prime }$. Singular values of zero are not retained in the solution.

Usage

SVD(X, tol = sqrt(.Machine$double.eps))

Arguments

X

a square symmetric matrix

tol

tolerance passed to QR

Value

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

See Also

svd, the standard svd function

Eigen

Examples

# NOT RUN {
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))
# }