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

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The duplication matrix   is the unique   matrix which, for any   symmetric matrix  , transforms   into  :

 .

For the   symmetric matrix  , this transformation reads

 


The explicit formula for calculating the duplication matrix for a   matrix is:

 

Where:

  •   is a unit vector of order   having the value   in the position   and 0 elsewhere;
  •   is a   matrix with 1 in position   and   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();
}

Elimination matrix

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An elimination matrix   is a   matrix which, for any   matrix  , transforms   into  :

 [1]

By the explicit (constructive) definition given by Magnus & Neudecker (1980), the   by   elimination matrix   is given by

 

where   is a unit vector whose  -th element is one and zeros elsewhere, and  .

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   matrix  , one choice for this transformation is given by

 .

Notes

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  1. ^ Magnus & Neudecker (1980), Definition 3.1

References

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  • 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