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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 n2×12n(n+1) matrix.

Arguments

n

a positive integer value for the order of the underlying matrix

Author

Frederick Novomestky fnovomes@poly.edu

Details

Let Ti,j be an n×n matrix with 1 in its (i,j) element 1i,jn. and zeroes elsewhere. These matrices are constructed by the function T.matrices. The formula for the transpose of matrix D is D=j=1ni=jnui,j(vecTi,j) where ui,j is the column vector in the order 12n(n+1) identity matrix for column k=(j1)n+i12j(j1). 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

Run this code
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 )

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