Duplication and elimination matrices

For matrices encoding row elimination operations, see Elementary matrix.

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 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):

$D_{n}\operatorname {vech} (A)=\operatorname {vec} (A)$.

For the 2 × 2 symmetric matrix $A=\left[{\begin{smallmatrix}a&b\\b&d\end{smallmatrix}}\right]$, this transformation reads

$D_{n}\operatorname {vech} (A)=\operatorname {vec} (A)\implies {\begin{bmatrix}1&0&0\\0&1&0\\0&1&0\\0&0&1\end{bmatrix}}{\begin{bmatrix}a\\b\\d\end{bmatrix}}={\begin{bmatrix}a\\b\\b\\d\end{bmatrix}}$

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

$D_{n}^{\mathrm {T} }=\sum _{i\geq j}u_{ij}(\operatorname {vec} T_{ij})^{\mathrm {T} }$

Where:

• u_ij is a unit vector of order ⁠1/2⁠n(n + 1) having the value 1 in the position (j − 1)n + i − ⁠1/2⁠j(j − 1) and 0 elsewhere;
• T_ij is a n × n matrix with 1 in position (i, j) and (j, i) and 0 elsewhere

Elimination matrix

An elimination matrix L_n is a ⁠n(n + 1)/2⁠ × n² matrix which, for any n × n matrix A, transforms vec(A) into vech(A):

$L_{n}\operatorname {vec} (A)=\operatorname {vech} (A).$

For the 2 × 2 matrix $A=\left[{\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right]$, one choice for this transformation is given by

$L_{n}\operatorname {vec} (A)=\operatorname {vech} (A)\implies {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}a\\c\\b\\d\end{bmatrix}}={\begin{bmatrix}a\\c\\d\end{bmatrix}}.$

References

• Magnus, Jan R.; Neudecker, Heinz (1980), "The elimination matrix: some lemmas and applications", SIAM Journal on Algebraic and Discrete Methods, 1 (4): 422–449, doi:10.1137/0601049, ISSN 0196-5212.
• Jan R. Magnus and Heinz Neudecker (1988), Matrix Differential Calculus with Applications in Statistics and Econometrics, Wiley. ISBN 0-471-98633-X.
• Jan R. Magnus (1988), Linear Structures, Oxford University Press. ISBN 0-19-520655-X