R: Duplication matrix

D.matrix {matrixcalc} | R Documentation

## 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)

### Arguments

`n` | a positive integer value for the order of the underlying matrix

### Details

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

### 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 )

[Package matrixcalc version 1.0-3 Index]