Duplication matrix

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 "× n(n+1)/2" matrix which, for any "n × n" symmetric matrix "A", transforms vech("A") into vec("A")::"D"n vech("A") = vec("A").}

Elimination matrix

The elimination matrix "L"_{n is the unique "n(n+1)/2 × n"² matrix which, for any "n × n" lower triangular matrix "A", transforms vec("A") into vech("A")::"L"n 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