| is.idempotent.matrix {matrixcalc} | R Documentation |
Test for idempotent square matrix
Description
This function returns a 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 FALSE value is returned.
Usage
is.idempotent.matrix(x, tol = 1e-08)
Arguments
x |
a numeric square matrix |
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
{\bf{y}} = {\bf{X}}\;{\bf{\beta }} + {\bf{e}} using the method of least squares.
{\bf{y}} is an order m random vector of dependent variables.
{\bf{X}} is an m \times n matrix whose columns are columns of
observations on one of the n - 1 independent variables. The first column
contains m ones. {\bf{e}} is an order m random vector of zero
mean residual values. {\bf{\beta }} is the order n vector of regression
parameters. The objective function that is minimized in the method of least squares is
\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
{\bf{\hat \beta }} = \left[ {\left( {{\bf{X'}}\;{\bf{X}}} \right)^{ - 1} \;{\bf{X'}}} \right]\;{\bf{y}}
The corresponding estimator for the residual vector is
{\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}}.
{\bf{M}} and {{\bf{X}}\;\left( {{\bf{X'}}\;{\bf{X}}} \right)^{ - 1} {\bf{X'}}} are idempotent.
Idempotency of {\bf{M}} enters into the estimation of the variance of the estimator.
Value
A TRUE or FALSE value.
Author(s)
Frederick Novomestky fnovomes@poly.edu
References
Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.
Chang, A. C., (1984). Fundamental Methods of Mathematical Economics, Third edition, McGraw-Hill.
Green, W. H. (2003). Econometric Analysis, Fifth edition, Prentice-Hall.
Horn, R. A. and C. R. Johnson (1990). Matrix Analysis, Cambridge University Press.
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 )