In mathematics, especially in linear algebra and matrix theory, the duplication matrix and the elimination matrix are linear transformations used for transforming half-vectorizationss of matrices into vectorizations or (respectively) vice versa.

Duplication matrix

symmetric matrix:

1. .

, this transformation reads

matrix is:

Where:

• and 0 elsewhere;

• and zero elsewhere

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

Elimination matrix

:

1. .

By the explicit (constructive) definition given by is given by

.

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

, one choice for this transformation is given by

1. .

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.

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