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\name{MatrixFactorization-class}
\title{Virtual Class "MatrixFactorization" of Matrix Factorizations}
%
\docType{class}
\keyword{algebra}
\keyword{array}
\keyword{classes}
%
\alias{MatrixFactorization-class}
%
\alias{BunchKaufmanFactorization-class}
\alias{CholeskyFactorization-class}
\alias{SchurFactorization-class}
\alias{LU-class}
\alias{QR-class}
%
\alias{determinant,MatrixFactorization,missing-method}
\alias{dim,MatrixFactorization-method}
\alias{dimnames,MatrixFactorization-method}
\alias{dimnames<-,MatrixFactorization,NULL-method}
\alias{dimnames<-,MatrixFactorization,list-method}
\alias{length,MatrixFactorization-method}
\alias{show,MatrixFactorization-method}
\alias{unname,MatrixFactorization-method}
%
\alias{show,BunchKaufmanFactorization-method}
\alias{show,CholeskyFactorization-method}
\alias{show,SchurFactorization-method}
\alias{show,LU-method}
\alias{show,QR-method}
%
\description{
\code{MatrixFactorization} is the virtual class of
factorizations of \eqn{m \times n}{m-by-n} matrices \eqn{A},
having the general form
\deqn{P_{1} A P_{2} = A_{1} \cdots A_{p}}{P1 * A * P2 = A1 * ... * Ap}
or (equivalently)
\deqn{A = P_{1}' A_{1} \cdots A_{p} P_{2}'}{A = P1' * A1 * ... * Ap * P2'}
where \eqn{P_{1}}{P1} and \eqn{P_{2}}{P2} are permutation matrices.
Factorizations requiring symmetric \eqn{A} have the constraint
\eqn{P_{2} = P_{1}'}{P2 = P1'}, and factorizations without row
or column pivoting have the constraints
\eqn{P_{1} = I_{m}}{P1 = Im} and \eqn{P_{2} = I_{n}}{P2 = In},
where \eqn{I_{m}}{Im} and \eqn{I_{n}}{In} are the
\eqn{m \times m}{m-by-m} and \eqn{n \times n}{n-by-n} identity matrices.
\code{CholeskyFactorization}, \code{BunchKaufmanFactorization},
\code{SchurFactorization}, \code{LU}, and \code{QR} are the virtual
subclasses of \code{MatrixFactorization} containing all Cholesky,
Bunch-Kaufman, Schur, LU, and QR factorizations, respectively.
}
\section{Slots}{
\describe{
\item{\code{Dim}}{an integer vector of length 2 giving the
dimensions of the factorized matrix.}
\item{\code{Dimnames}}{a list of length 2 preserving the
\code{dimnames} of the factorized matrix. Each element
must be \code{NULL} or a character vector of length equal
to the corresponding element of \code{Dim}.}
}
}
\section{Methods}{
\describe{
\item{\code{determinant}}{\code{signature(x = "MatrixFactorization", logarithm = "missing")}:
sets \code{logarithm = TRUE} and recalls the generic function.}
\item{\code{dim}}{\code{signature(x = "MatrixFactorization")}:
returns \code{x@Dim}.}
\item{\code{dimnames}}{\code{signature(x = "MatrixFactorization")}:
returns \code{x@Dimnames}.}
\item{\code{dimnames<-}}{\code{signature(x = "MatrixFactorization", value = "NULL")}:
returns \code{x} with \code{x@Dimnames} set to \code{list(NULL, NULL)}.}
\item{\code{dimnames<-}}{\code{signature(x = "MatrixFactorization", value = "list")}:
returns \code{x} with \code{x@Dimnames} set to \code{value}.}
\item{\code{length}}{\code{signature(x = "MatrixFactorization")}:
returns \code{prod(x@Dim)}.}
\item{\code{show}}{\code{signature(object = "MatrixFactorization")}:
prints the internal representation of the factorization using
\code{\link{str}}.}
\item{\code{solve}}{\code{signature(a = "MatrixFactorization", b = .)}:
see \code{\link{solve-methods}}.}
\item{\code{unname}}{\code{signature(obj = "MatrixFactorization")}:
returns \code{obj} with \code{obj@Dimnames} set to
\code{list(NULL, NULL)}.}
}
}
\seealso{
Classes extending \code{CholeskyFactorization}, namely
\code{\linkS4class{Cholesky}}, \code{\linkS4class{pCholesky}},
and \code{\linkS4class{CHMfactor}}.
Classes extending \code{BunchKaufmanFactorization}, namely
\code{\linkS4class{BunchKaufman}} and \code{\linkS4class{pBunchKaufman}}.
Classes extending \code{SchurFactorization}, namely
\code{\linkS4class{Schur}}.
Classes extending \code{LU}, namely
\code{\linkS4class{denseLU}} and \code{\linkS4class{sparseLU}}.
Classes extending \code{QR}, namely \code{\linkS4class{sparseQR}}.
Generic functions \code{\link{Cholesky}}, \code{\link{BunchKaufman}},
\code{\link{Schur}}, \code{\link{lu}}, and \code{\link{qr}} for
\emph{computing} factorizations.
Generic functions \code{\link{expand1}} and \code{\link{expand2}}
for constructing matrix factors from \code{MatrixFactorization}
objects.
}
\examples{
showClass("MatrixFactorization")
}
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