Duplication matrix

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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.

[edit] Duplication matrix

The duplication matrix D_n is the unique n² × 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).

[edit] Elimination matrix

The elimination matrix L_n is the unique n(n+1)/2 × n² matrix which, for any n × n lower triangular matrix A, transforms vec(A) into vech(A):

L_n vec(A) = vech(A).

[edit] 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 0-85264-299-X.