Duplication and elimination matrices

The duplication matrix $D\_n$ is the unique $n^2\; \backslash times\; \backslash frac\{n(n+1)\}\{2\}$ matrix which, for any $n\; \backslash times\; n$ symmetric matrix $A$, transforms $\backslash mathrm\{vech\}(A)$ into $\backslash mathrm\{vec\}(A)$:

:$D\_n\; \backslash mathrm\{vech\}(A)\; =\; \backslash mathrm\{vec\}(A)$.

For the $2\; \backslash times\; 2$ symmetric matrix , this transformation reads

:

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

$D^T\_n\; =\; \backslash sum\; \backslash limits\_\{i\; \backslash ge\; j\}\; u\_\{ij\}\; (\backslash mathrm\{vec\}T\_\{ij\})^T$

Where:

• $u\_\{ij\}$ is a unit vector of order $\backslash frac\{1\}\{2\}\; n\; (n+1)$ having the value $1$ in the position $(j-1)n+i\; -\; \backslash frac\{1\}\{2\}j(j-1)$ and 0 elsewhere;
• $T\_\{ij\}$ is a $n\; \backslash times\; n$ 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) {

arma::mat out((n*(n+1))/2, n*n, arma::fill::zeros);

for (int j = 0; j < n; ++j) {

for (int i = j; i < n; ++i) {

arma::vec u((n*(n+1))/2, arma::fill::zeros);

u(j*n+i-((j+1)*j)/2) = 1.0;

arma::mat T(n,n, arma::fill::zeros);

T(i,j) = 1.0;

T(j,i) = 1.0;

out += u * arma::trans(arma::vectorise(T));

}

}

return out.t();

}

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

:$L\_n\; \backslash mathrm\{vec\}(A)\; =\; \backslash mathrm\{vech\}(A)$. {{harvtxt|Magnus|Neudecker|1980}}, Definition 3.1

By the explicit (constructive) definition given by {{harvtxt|Magnus|Neudecker|1980}}, the $\backslash frac\{1\}\{2\}n(n+1)$ by $n^2$ elimination matrix $L\_n$ is given by

:$L\_n\; =\; \backslash sum\_\{i\; \backslash geq\; j\}\; u\_\{ij\}\; \backslash mathrm\{vec\}(E\_\{ij\})^T\; =\; \backslash sum\_\{i\; \backslash geq\; j\}\; (u\_\{ij\}\backslash otimes\; e\_j^T\; \backslash otimes\; e\_i^T),$

where $e\_i$ is a unit vector whose $i$-th element is one and zeros elsewhere, and $E\_\{ij\}\; =\; e\_ie\_j^T$.

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

arma::mat elimination_matrix(const int &n) {

arma::mat out((n*(n+1))/2, n*n, arma::fill::zeros);

for (int j = 0; j < n; ++j) {

arma::rowvec e_j(n, arma::fill::zeros);

e_j(j) = 1.0;

for (int i = j; i < n; ++i) {

arma::vec u((n*(n+1))/2, arma::fill::zeros);

u(j*n+i-((j+1)*j)/2) = 1.0;

arma::rowvec e_i(n, arma::fill::zeros);

e_i(i) = 1.0;

out += arma::kron(u, arma::kron(e_j, e_i));

}

}

return out;

}

For the $2\; \backslash times\; 2$ matrix , one choice for this transformation is given by

: .

• {{Citation | last1=Magnus | first1=Jan R. | last2=Neudecker | first2=Heinz | title=The elimination matrix: some lemmas and applications | doi=10.1137/0601049 | year=1980 | journal= SIAM Journal on Algebraic and Discrete Methods| issn=0196-5212 | volume=1 | issue=4 | pages=422–449| url=http://repository.uvt.nl/id/ir-uvt-nl:oai:wo.uvt.nl:153210 }}.
• 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}}

Category:Matrices (mathematics)

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