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 formula_1 is the unique formula_2 matrix which, for any formula_3 symmetric matrix formula_4, transforms formula_5 into formula_6:
For the formula_8 symmetric matrix formula_9, this transformation reads
The explicit formula for calculating the duplication matrix for a formula_11 matrix is: 
formula_12
Where: 
Elimination matrix.
An elimination matrix formula_21 is a formula_22 matrix which, for any formula_11 matrix formula_24, transforms formula_6 into formula_5:
For the formula_28 matrix formula_29, one choice for this transformation is given by