matrixcalc (version 1.0-6)

elimination.matrix: 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)

Value

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

Arguments

n

row or column dimension

Author

Frederick Novomestky

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 \({\bf{L}} = \sum\limits_{j = 1}^n {\sum\limits_{i = j}^n {{{\bf{u}}_{i,j}}{{\left( {vec\;{{\bf{E}}_{i,j}}} \right)}^\prime }} } \) \({{{\bf{u}}_{i,j}}}\) are the order \(n\left( {n + 1} \right)/2\) vectors constructed by the function u.vectors. \({{{\bf{E}}_{i,j}}}\) are the \( n \times n\) matrices constructed by the function E.matrices.

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 )