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 ${\displaystyle D_{n}}$ is the unique ${\displaystyle n^{2}\times {\frac {n(n+1)}{2}}}$ matrix which, for any ${\displaystyle n\times n}$ symmetric matrix ${\displaystyle A}$, transforms ${\displaystyle \mathrm {vech} (A)}$ into ${\displaystyle \mathrm {vec} (A)}$:

${\displaystyle D_{n}\mathrm {vech} (A)=\mathrm {vec} (A)}$.

For the ${\displaystyle 2\times 2}$ symmetric matrix ${\displaystyle A=\left[{\begin{smallmatrix}a&b\\b&d\end{smallmatrix}}\right]}$, this transformation reads

${\displaystyle D_{n}\mathrm {vech} (A)=\mathrm {vec} (A)\implies {\begin{bmatrix}1&0&0\\0&1&0\\0&1&0\\0&0&1\end{bmatrix}}{\begin{bmatrix}a\\b\\d\end{bmatrix}}={\begin{bmatrix}a\\b\\b\\d\end{bmatrix}}}$

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

${\displaystyle D_{n}^{T}=\sum \limits _{i\geq j}u_{ij}(\mathrm {vec} T_{ij})^{T}}$

Where:

• ${\displaystyle u_{ij}}$ is a unit vector of order ${\displaystyle {\frac {1}{2}}n(n+1)}$ having the value ${\displaystyle 1}$ in the position ${\displaystyle (j-1)n+i-{\frac {1}{2}}j(j-1)}$ and 0 elsewhere;
• ${\displaystyle T_{ij}}$ is a ${\displaystyle n\times n}$ matrix with 1 in position ${\displaystyle (i,j)}$ and ${\displaystyle (j,i)}$ 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();
}

Elimination matrix

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

${\displaystyle L_{n}\mathrm {vec} (A)=\mathrm {vech} (A)}$. ^[1]

By the explicit (constructive) definition given by Magnus & Neudecker (1980), the ${\displaystyle {\frac {1}{2}}n(n+1)}$ by ${\displaystyle n^{2}}$ elimination matrix ${\displaystyle L_{n}}$ is given by

${\displaystyle L_{n}=\sum _{i\geq j}u_{ij}\mathrm {vec} (E_{ij})^{T}=\sum _{i\geq j}(u_{ij}\otimes e_{j}^{T}\otimes e_{i}^{T}),}$

where ${\displaystyle e_{i}}$ is a unit vector whose ${\displaystyle i}$-th element is one and zeros elsewhere, and ${\displaystyle E_{ij}=e_{i}e_{j}^{T}}$.

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 ${\displaystyle 2\times 2}$ matrix ${\displaystyle A=\left[{\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right]}$, one choice for this transformation is given by

${\displaystyle L_{n}\mathrm {vec} (A)=\mathrm {vech} (A)\implies {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}a\\c\\b\\d\end{bmatrix}}={\begin{bmatrix}a\\c\\d\end{bmatrix}}}$.

Notes

References

• 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