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 Dn is the unique n2 × n(n+1)/2 matrix which, for any n × n symmetric matrix A, transforms vech(A) into vec(A):
- Dn vech(A) = vec(A).
Elimination matrix
The elimination matrix Ln is the unique n(n+1)/2 × n2 matrix which, for any n × n lower triangular matrix A, transforms vec(A) into vech(A):
- Ln vec(A) = vech(A).
References
- 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 019520655X
