Duplication and elimination matrices

In mathematics, especially in linear algebra and matrix theory, the duplication matrix is used for transforming the half-vectorization of a symmetric matrix into its vectorization. Specifically, 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).

Here vec(A) is the n² × 1 column vector obtain by stacking the columns of the matrix A on top of one another:

vec(A) = [ A_1,1, ..., A_n,1, A_1,2, ..., A_n,2, ..., A_1,n, ..., A_n,n ]^T.

For A symmetric, vech(A), the half-vectorization of A, is the n(n+1)/2 × 1 column vector obtained by vectorizing only the lower triangular portion of A:

vech(A) = [ A_1,1, ..., A_m,1, A_2,2, ..., A_n,2, ..., A_n-1,n-1,A_n-1,n, A_n,n ]^T.

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