Duplication and elimination matrices

Last updated December 05, 2021

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 $\mathrm {vech} (A)$ into $\mathrm {vec} (A)$ :

D_{n}\mathrm {vech} (A)=\mathrm {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}\mathrm {vech} (A)=\mathrm {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:

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

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 $\mathrm {vec} (A)$ into $\mathrm {vech} (A)$ :

L_{n}\mathrm {vec} (A)=\mathrm {vech} (A)

. ^[1]

By the explicit (constructive) definition given by Magnus & Neudecker (1980), the ${\frac {1}{2}}n(n+1)$ by $n^{2}$ elimination matrix $L_{n}$ is given by

L_{n}=\sum _{i\geq j}u_{ij}\mathrm {vec} (E_{ij})^{T}=\sum _{i\geq j}(u_{ij}\otimes e_{j}^{T}\otimes e_{i}^{T}),

where $e_{i}$ is a unit vector whose $i$ -th element is one and zeros elsewhere, and $E_{ij}=e_{i}e_{j}^{T}$ .

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}\mathrm {vec} (A)=\mathrm {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}}

.

Notes

↑ Magnus & Neudecker (1980), Definition 3.1

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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

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[1] Magnus & Neudecker (1980), Definition 3.1

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