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.
is.idempotent.matrix(x, tol = 1e-08)A TRUE or FALSE value.
a numeric square matrix
a numeric tolerance level usually left out
Frederick Novomestky fnovomes@poly.edu
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.
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.
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 )
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