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\name{is.idempotent.matrix}
\alias{is.idempotent.matrix}
\title{ Test for idempotent square matrix }
\description{
This function returns a \code{TRUE} value if the square matrix argument x
is idempotent, that is, the product of the matrix with itself is the matrix.
The equality test is performed to within the specified tolerance level. If
the matrix is not idempotent, then a \code{FALSE} value is returned.
}
\usage{
is.idempotent.matrix(x, tol = 1e-08)
}
\arguments{
\item{x}{ a numeric square matrix }
\item{tol}{ a numeric tolerance level usually left out }
}
\details{
Idempotent matrices are used in econometric analysis. Consider the problem of
estimating the regression parameters of a standard linear model
\eqn{{\bf{y}} = {\bf{X}}\;{\bf{\beta }} + {\bf{e}}} using the method of least squares.
\eqn{{\bf{y}}} is an order \eqn{m} random vector of dependent variables.
\eqn{{\bf{X}}} is an \eqn{m \times n} matrix whose columns are columns of
observations on one of the \eqn{ n - 1} independent variables. The first column
contains \eqn{m} ones. \eqn{{\bf{e}}} is an order \eqn{m} random vector of zero
mean residual values. \eqn{{\bf{\beta }}} is the order \eqn{n} vector of regression
parameters. The objective function that is minimized in the method of least squares is
\eqn{\left( {{\bf{y}} - {\bf{X}}\;{\bf{\beta }}} \right)^\prime \left( {{\bf{y}} - {\bf{X}}\;{\bf{\beta }}} \right)}.
The solution to ths quadratic programming problem is
\eqn{{\bf{\hat \beta }} = \left[ {\left( {{\bf{X'}}\;{\bf{X}}} \right)^{ - 1} \;{\bf{X'}}} \right]\;{\bf{y}}}
The corresponding estimator for the residual vector is
\eqn{{\bf{\hat e}} = {\bf{y}} - {\bf{X}}\;{\bf{\hat \beta }} = \left[ {{\bf{I}} - {\bf{X}}\;\left( {{\bf{X'}}\;{\bf{X}}} \right)^{ - 1} {\bf{X'}}} \right]{\bf{y}} = {\bf{M}}\;{\bf{y}}}.
\eqn{{\bf{M}}} and \eqn{{{\bf{X}}\;\left( {{\bf{X'}}\;{\bf{X}}} \right)^{ - 1} {\bf{X'}}}} are idempotent.
Idempotency of \eqn{{\bf{M}}} enters into the estimation of the variance of the estimator.
}
\value{
A TRUE or FALSE value.
}
\references{
Bellman, R. (1987). \emph{Matrix Analysis}, Second edition, Classics in Applied Mathematics,
Society for Industrial and Applied Mathematics.
Chang, A. C., (1984). \emph{Fundamental Methods of Mathematical Economics},
Third edition, McGraw-Hill.
Green, W. H. (2003). \emph{Econometric Analysis}, Fifth edition, Prentice-Hall.
Horn, R. A. and C. R. Johnson (1990). \emph{Matrix Analysis}, Cambridge University Press.
}
\author{ Frederick Novomestky \email{fnovomes@poly.edu} }
\examples{
A <- diag( 1, 3 )
is.idempotent.matrix( A )
B <- matrix( c( 1, 2, 3, 4 ), nrow=2, byrow=TRUE )
is.idempotent.matrix( B )
C <- matrix( c( 1, 0, 0, 0 ), nrow=2, byrow=TRUE )
is.idempotent.matrix( C )
}
\keyword{ math }
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