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 vice versa.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
An elimination matrix is a matrix which, for any matrix, transforms into :
For the matrix, one choice for this transformation is given by
