Duplication and elimination matrices

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 is the unique matrix which, for any symmetric matrix , transforms into :

For the symmetric matrix , this transformation reads

The explicit formula for calculating the duplication matrix for a matrix is:

Where:

• is a unit vector of order having the value in the position and 0 elsewhere;
• is a matrix with 1 in position and and zero elsewhere

Here is a C++ function using Armadillo (C++ library):

Elimination matrix

An elimination matrix is a matrix which, for any matrix , transforms into :

By the explicit (constructive) definition given by , the by elimination matrix is given by

where is a unit vector whose -th element is one and zeros elsewhere, and .

Here is a C++ function using Armadillo (C++ library):

For the matrix , one choice for this transformation is given by

Notes

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.