duplication.matrix {matrixcalc}

Duplication matrix for n by n matrices

Description

This function returns a matrix with n * n rows and n * ( n + 1 ) / 2 columns that transforms vech(A) to vec(A) where A is a symmetric n by n matrix.

Usage

duplication.matrix(n=1)

Arguments

n

Row and column dimension

Details

This function is a wrapper function for the function D.matrix. Let {\bf{T}}_{i,j} be an n \times n matrix with 1 in its \left( {i,j} \right) element 1 \le i,j \le n. and zeroes elsewhere. These matrices are constructed by the function T.matrices. The formula for the transpose of matrix \bf{D} is {\bf{D'}} = \sum\limits_{j = 1}^n {\sum\limits_{i = j}^n {{{\bf{u}}_{i,j}}\;{{\left( {vec\;{{\bf{T}}_{i,j}}} \right)}^\prime }} } where {{{\bf{u}}_{i,j}}} is the column vector in the order \frac{1}{2}n\left( {n + 1} \right) identity matrix for column k = \left( {j - 1} \right)n + i - \frac{1}{2}j\left( {j - 1} \right). The function u.vectors generates these vectors.

Value

It returns an {n^2}\; \times \;\frac{1}{2}n\left( {n + 1} \right) matrix.

Author(s)

Frederick Novomestky [e-mail], Kurt Hornik [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

D.matrix, vec, vech

Examples

D <- duplication.matrix( 3 )
A <- matrix( c( 1, 2, 3,
                2, 3, 4,
                3, 4, 5), nrow=3, byrow=TRUE )
vecA <- vec( A )
vechA<- vech( A )
y <- D %*% vechA
print( y )
print( vecA )