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
![{\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}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The explicit formula for calculating the duplication matrix for a
matrix is:
![{\displaystyle D_{n}^{T}=\sum \limits _{i\geq j}u_{ij}(\mathrm {vec} T_{ij})^{T}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
:
. [1]
By the explicit (constructive) definition given by Magnus & Neudecker (1980), the
by
elimination matrix
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}),}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
- ^ Magnus & Neudecker (1980), Definition 3.1
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