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

The elimination matrix L_n is the unique 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 = $\left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right]$, this transformation reads

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

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