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 = ${\begin{bmatrix}a&b\\b&d\end{bmatrix}}$, 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 lower triangular matrix A, transforms vec(A) into vech(A):}

L_n vec(A) = vech(A).

For the 2×2 lower triangular matrix A = ${\begin{bmatrix}a&0\\b&d\end{bmatrix}}$, this transformation reads

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

References

• 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 019520655X