Elimination matrix for lower triangular matrices
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)
elimination.matrix(n)
n |
row or column dimension |
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.
An ≤ft[ {\frac{1}{2}n≤ft( {n + 1} \right)} \right] \times {n^2} matrix.
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.
Frederick Novomestky fnovomes@poly.edu
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.
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 )
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