Inmathematics, especially inlinear algebra andmatrix theory, theduplication matrix and theelimination matrix arelinear transformations used for transforminghalf-vectorizations ofmatrices intovectorizations or (respectively) vice versa.

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${\displaystyle \mathrm{v}\mathrm{e}\mathrm{c}\mathrm{h}(A)}$ into${\displaystyle \mathrm{v}\mathrm{e}\mathrm{c}(A)}$:

${\displaystyle D_n\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{h}(A)=\mathrm{v}\mathrm{e}\mathrm{c}(A)}$.

For the${\displaystyle 2\times 2}$ symmetric matrix, this transformation reads

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

${\displaystyle D_n^T=\sum _{i\ge j}{u}_{ij}(\mathrm{v}\mathrm{e}\mathrm{c}{T}_{ij}{)}^T}$

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 aC++ function usingArmadillo (C++ library):

arma::matduplication_matrix(constint&n){arma::matout((n*(n+1))/2,n*n,arma::fill::zeros);for(intj=0;j<n;++j){for(inti=j;i<n;++i){arma::vecu((n*(n+1))/2,arma::fill::zeros);u(j*n+i-((j+1)*j)/2)=1.0;arma::matT(n,n,arma::fill::zeros);T(i,j)=1.0;T(j,i)=1.0;out+=u*arma::trans(arma::vectorise(T));}}returnout.t();}

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${\displaystyle \mathrm{v}\mathrm{e}\mathrm{c}(A)}$ into${\displaystyle \mathrm{v}\mathrm{e}\mathrm{c}\mathrm{h}(A)}$:

${\displaystyle L_n\mathrm{v}\mathrm{e}\mathrm{c}(A)=\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{h}(A)}$. ^[1]

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

${\displaystyle L_n=\sum _{i\ge j}{u}_{ij}\mathrm{v}\mathrm{e}\mathrm{c}(E_{ij}{)}^T=\sum _{i\ge j}(u_{ij}\otimes e_j^T\otimes e_i^T),}$

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 aC++ function usingArmadillo (C++ library):

arma::matelimination_matrix(constint&n){arma::matout((n*(n+1))/2,n*n,arma::fill::zeros);for(intj=0;j<n;++j){arma::rowvece_j(n,arma::fill::zeros);e_j(j)=1.0;for(inti=j;i<n;++i){arma::vecu((n*(n+1))/2,arma::fill::zeros);u(j*n+i-((j+1)*j)/2)=1.0;arma::rowvece_i(n,arma::fill::zeros);e_i(i)=1.0;out+=arma::kron(u,arma::kron(e_j,e_i));}}returnout;}

For the${\displaystyle 2\times 2}$ matrix, one choice for this transformation is given by

.

• 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 978-0-471-98633-1.
• Jan R. Magnus (1988),Linear Structures, Oxford University Press.ISBN 978-0-19-520655-5

• Matrices (mathematics)