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 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 An elimination matrix is a matrix which, for any matrix , transforms into : . , Definition 3.1

By the explicit (constructive) definition given by , 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

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.

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