Duplication Matrix

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 $n^2 \times \frac$ matrix which, for any $n \times n$ symmetric matrix $A$, transforms $\mathrm(A)$ into $\mathrm(A)$:
:$D_n \mathrm(A) = \mathrm(A)$.

For the $2 \times 2$ symmetric matrix A=\left begin a & b \\ b & d \end\right/math>, this transformation reads
: $D_n \mathrm(A) = \mathrm(A) \implies \begin 1&0&0 \\ 0&1&0 \\ 0&1&0 \\ 0&0&1 \end \begin a \\ b \\ d \end = \begin a \\ b \\ b \\ d \end$

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

$D^T_n = \sum \limits_ u_ (\mathrmT_)^T$

Where:
*$u_$ is a unit vector of order $\frac n (n+1)$ having the value $1$ in the position $(j-1)n+i - \fracj(j-1)$ and 0 elsewhere;
*$T_$ is a $n \times n$ matrix with 1 in position $(i,j)$ and $(j,i)$ and zero elsewhere

Here is a C++ function using Armadillo (C++ library):

arma::mat duplication_matrix(const int &n)

Elimination matrix

An elimination matrix $L_n$ is a $\frac \times n^2$ matrix which, for any $n \times n$ matrix $A$, transforms $\mathrm(A)$ into $\mathrm(A)$:
:$L_n \mathrm(A) = \mathrm(A)$. , Definition 3.1

By the explicit (constructive) definition given by , the $\fracn(n+1)$ by $n^2$ elimination matrix $L_n$ is given by
:$L_n = \sum_ u_ \mathrm(E_)^T = \sum_ (u_\otimes e_j^T \otimes e_i^T),$
where $e_i$ is a unit vector whose $i$-th element is one and zeros elsewhere, and $E_ = e_ie_j^T$.

Here is a C++ function using Armadillo (C++ library):

arma::mat elimination_matrix(const int &n)

For the $2 \times 2$ matrix A = \left begin a & b \\ c & d \end\right/math>, one choice for this transformation is given by
: $L_n \mathrm(A) = \mathrm(A) \implies \begin 1&0&0&0 \\ 0&1&0&0 \\ 0&0&0&1 \end \begin a \\ c \\ b \\ d \end = \begin a \\ c \\ d \end$.

Notes

References

*.
*Jan R. Magnus and Heinz Neudecker (1988), ''Matrix Differential Calculus with Applications in Statistics and Econometrics'', Wiley. .
*Jan R. Magnus (1988), ''Linear Structures'', Oxford University Press. {{ISBN, 0-19-520655-X

Matrices

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