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 $n^{2}\times {\frac {n(n+1)}{2}}$ matrix which, for any $n\times n$ symmetric matrix $A$, transforms $vech(A)$ into $vec(A)$:

$D_{n}vech(A)=vec(A)$.

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

$D_{n}vech(A)=vec(A)\implies {\begin{bmatrix}1&0&0\\0&1&0\\0&1&0\\0&0&1\end{bmatrix}}{\begin{bmatrix}a\\b\\d\end{bmatrix}}={\begin{bmatrix}a\\b\\b\\d\end{bmatrix}}$

The explicit formula for calculating the duplication matrix for a $n\times n$ matrix is:

$D_{n}^{T}=\sum \limits _{i\geq j}u_{ij}(vecT_{ij})^{T}$

Where:

• $u_{ij}$ is a unit vector of order ${\frac {1}{2}}n(n+1)$ having the value $1$ in the position $(j-1)n+i-{\frac {1}{2}}j(j-1)$ and 0 elsewhere;
• $T_{ij}$ is a $n\times n$ matrix with 1 in position $(i,j)$ and $(j,i)$ and zero elsewhere

Elimination matrix

An elimination matrix $L_{n}$ is a ${\frac {n(n+1)}{2}}\times n^{2}$ matrix which, for any $n\times n$ matrix $A$, transforms $vec(A)$ into $vech(A)$:

$L_{n}vec(A)=vech(A)$. 

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

$L_{n}vec(A)=vech(A)\implies {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}a\\c\\b\\d\end{bmatrix}}={\begin{bmatrix}a\\c\\d\end{bmatrix}}$.

References

• Magnus, Jan R.; Neudecker, Heinz (1980), "The elimination matrix: some lemmas and applications", SIAM 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