elimination.matrix: Elimination matrix for lower triangular matrices

In matrixcalc: Collection of Functions for Matrix Calculations

Elimination matrix for lower triangular matrices

Description

This function returns a matrix with n * ( n + 1 ) / 2 rows and N * n columns which for any lower triangular matrix A transforms vec( A ) into vech(A)

Usage

elimination.matrix(n)

Arguments

n	row or column dimension

Details

This function is a wrapper function to the function L.matrix. The formula used to compute the L matrix which is also called the elimination matrix is {\bf{L}} = ∑\limits_{j = 1}^n {∑\limits_{i = j}^n {{{\bf{u}}_{i,j}}{{≤ft( {vec\;{{\bf{E}}_{i,j}}} \right)}^\prime }} } {{{\bf{u}}_{i,j}}} are the order n≤ft( {n + 1} \right)/2 vectors constructed by the function u.vectors. {{{\bf{E}}_{i,j}}} are the n \times n matrices constructed by the function E.matrices.

Value

An ≤ft[ {\frac{1}{2}n≤ft( {n + 1} \right)} \right] \times {n^2} matrix.

Note

If the argument is not an integer, the function displays an error message and stops. If the argument is less than two, the function displays an error message and stops.

Author(s)

Frederick Novomestky [e-mail]

References

Magnus, J. R. and H. Neudecker (1980). The elimination matrix, some lemmas and applications, SIAM Journal on Algebraic Discrete Methods, 1(4), December 1980, 422-449.

Magnus, J. R. and H. Neudecker (1999) Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

See Also

E.matrices, L.matrix, u.vectors

Examples

L <- elimination.matrix( 4 )
A <- lower.triangle( matrix( seq( 1, 16, 1 ), nrow=4, byrow=TRUE ) )
vecA <- vec( A )
vechA <- vech( A )
y <- L %*% vecA
print( y )
print( vechA )

matrixcalc documentation built on Sept. 15, 2022, 1:05 a.m.