dcap: The Duplication Matrix

In jeksterslab/linearAlgebra: Linear Algebra with R

Description

Creates a duplication matrix.

Usage

dcap(k)

Arguments

k	Positive integer. Dimension of the k by k matrix.

Details

The duplication matrix \mathbf{D}_{k} is the k^2 \times \frac{1}{2} k ≤ft( k + 1 \right) matrix for a given k \times k symmetric matrix \mathbf{A} where

\mathbf{D}_{k} \mathrm{vech} ≤ft( \mathbf{A} \right) = \mathrm{vec} ≤ft( \mathbf{A} \right) \quad ≤ft( \mathbf{A} = \mathbf{A}^{\prime} \right)

\mathrm{vec} ≤ft( \cdot \right) is the vectorization of a matrix, and \mathrm{vech} ≤ft( \cdot \right) is the half-vectorization of a matrix.

Value

A matrix.

Author(s)

Ivan Jacob Agaloos Pesigan

References

Wikipedia: Duplication matrix

Wikipedia: Vectorization

Magnus, J. R., & Neudecker, H. (1980). The elimination matrix: Some lemmas and applications. SIAM Journal on Algebraic Discrete Methods, 1(4), 422–449. https://doi.org/10.1137/0601049

Magnus, J. R., & Neudecker, H. (2019). Matrix differential calculus with applications in statistics and econometrics. Wiley. https://doi.org/10.1002/9781119541219

See Also

Other Symmetric Functions: mcap_cor(), mcap_diag(), mcap_sym(), pinv_of_dcap(), sym_of_vechs(), sym_of_vech()

Examples

dcap(3)

jeksterslab/linearAlgebra documentation built on Dec. 20, 2021, 10:10 p.m.