matrixcalc (version 1.0-6)

D.matrix: Duplication matrix

Description

This function constructs the linear transformation D that maps vech(A) to vec(A) when A is a symmetric matrix

Usage

D.matrix(n)

Value

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

Arguments

n

a positive integer value for the order of the underlying matrix

Author

Frederick Novomestky

Details

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.

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

T.matrices, u.vectors

Examples

D <- D.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 )