Duplication and elimination matrices

In mathematics, especially in linear algebra and matrix theory, the duplication matrix and the elimination matrix are linear transformations used for transforming half-vectorizations of matrices into vectorizations or (respectively) vice versa.

Duplication matrix

The duplication matrix D_n is the unique n² × n(n+1)/2 matrix which, for any n × n symmetric matrix A, transforms vech(A) into vec(A):

D_n vech(A) = vec(A).

For the 2×2 symmetric matrix A = $\left[{\begin{smallmatrix}a&b\\b&d\end{smallmatrix}}\right]$, this transformation reads

${\begin{bmatrix}1&0&0\\0&1&0\\0&1&0\\0&0&1\end{bmatrix}}{\begin{bmatrix}a\\b\\d\end{bmatrix}}={\begin{bmatrix}a\\b\\b\\d\end{bmatrix}}$

Elimination matrix

An elimination matrix L_n is a n(n+1)/2 × n² matrix which, for any n × n matrix A, transforms vec(A) into vech(A):

L_n vec(A) = vech(A). ^[1]

For the 2×2 matrix A = $\left[{\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right]$, one choice for this transformation is given by

${\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}a\\c\\b\\d\end{bmatrix}}={\begin{bmatrix}a\\c\\d\end{bmatrix}}$.

Notes

1. Magnus & Neudecker (1980), Definition 3.1

References

• Magnus, Jan R.; Neudecker, Heinz (1980), "The elimination matrix: some lemmas and applications", Society for Industrial and Applied Mathematics. Journal on Algebraic and Discrete Methods, 1 (4): 422–449, doi:10.1137/0601049, ISSN 0196-5212.
• Jan R. Magnus and Heinz Neudecker (1988), Matrix Differential Calculus with Applications in Statistics and Econometrics, Wiley. ISBN 0-471-98633-X.
• Jan R. Magnus (1988), Linear Structures, Oxford University Press. ISBN 0-19-520655-X