duplication.matrix: Duplication matrix for n by n matrices

Description Usage Arguments Details Value Author(s) References See Also Examples

View source: R/duplication.matrix.R

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

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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 ≤ft( {i,j} \right) element 1 ≤ i,j ≤ 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'}} = ∑\limits_{j = 1}^n {∑\limits_{i = j}^n {{{\bf{u}}_{i,j}}\;{{≤ft( {vec\;{{\bf{T}}_{i,j}}} \right)}^\prime }} } where {{{\bf{u}}_{i,j}}} is the column vector in the order \frac{1}{2}n≤ft( {n + 1} \right) identity matrix for column k = ≤ft( {j - 1} \right)n + i - \frac{1}{2}j≤ft( {j - 1} \right). The function u.vectors generates these vectors.

Value

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

Author(s)

Frederick Novomestky fnovomes@poly.edu, Kurt Hornik Kurt.Hornik@wu-wien.ac.at

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

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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 )

Example output

      [,1]
 [1,]    1
 [2,]    2
 [3,]    3
 [4,]    2
 [5,]    3
 [6,]    4
 [7,]    3
 [8,]    4
 [9,]    5
      [,1]
 [1,]    1
 [2,]    2
 [3,]    3
 [4,]    2
 [5,]    3
 [6,]    4
 [7,]    3
 [8,]    4
 [9,]    5

matrixcalc documentation built on May 2, 2019, 1:45 p.m.