Duplication and elimination matrices - Alchetron, the free social encyclopedia

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 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).

For the 2×2 symmetric matrix A = [ a b b d ] , this transformation reads

[ 1 0 0 0 1 0 0 1 0 0 0 1 ] [ a b d ] = [ a b b d ]

Elimination matrix

An elimination matrix L_n is a n(n+1)/2 × n² matrix which, for any n × n matrix A, transforms vec(A) into vech(A):

L_n vec(A) = vech(A).

For the 2×2 matrix A = [ a b c d ] , one choice for this transformation is given by

[ 1 0 0 0 0 1 0 0 0 0 0 1 ] [ a c b d ] = [ a c d ] .

References

Duplication and elimination matrices Wikipedia

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Contents

Duplication matrix

Elimination matrix

References