Duplication and elimination matrices explained

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

D _{n} is the unique n ^{2} x (n(n +1 ))/(2) matrix which, for any n x n symmetric matrix A , transforms vech (A) into vec (A) :D _{n} vech (A) = vec (A) .

For the

2 x 2 symmetric matrix A = \left[\begin{smallmatrix} a & b \ b & d \end{smallmatrix}\right] , this transformation readsD _{n} vech (A) = 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

n x n matrix is: D ^{T} _{n} = \sum \limits _{i} u _{ij} ( vec T _{ij} ) ^{T}

Where:

u _{ij} is a unit vector of order (1)/(2) n (n +1 ) having the value 1 in the position (j -1 )n + i - (1)/(2) j(j -1 ) and 0 elsewhere; T _{ij} is a n x n matrix with 1 in position (i,j) and (j,i) and zero elsewhere

Here is a C++ function using Armadillo (C++ library):arma::mat duplication_matrix(const int &n)

Elimination matrix

An elimination matrix

L _{n} is a (n(n +1 ))/(2) x n ^{2} matrix which, for any n x n matrix A , transforms vec (A) into vech (A) :L _{n} vec (A) = vech (A) . ^[1]

By the explicit (constructive) definition given by, the

(1)/(2) n(n +1 ) by n ^{2} elimination matrix L _{n} is given byL _{n} = \sum _{i} u _{ij} vec (E _{ij} ) ^{T} = \sum _{i} (u _{ij} ⊗ e ^{T} _{j} ⊗ e ^{T),} _{i} where e _{i} is a unit vector whose i -th element is one and zeros elsewhere, and E _{ij} = e _{ie} ^{T} ^{j} .

Here is a C++ function using Armadillo (C++ library):arma::mat elimination_matrix(const int &n)

For the

2 x 2 matrix A = \left[\begin{smallmatrix} a & b \ c & d \end{smallmatrix}\right] , one choice for this transformation is given byL _{n} vec (A) = 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

1. , Definition 3.1

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.