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 n \( {\displaystyle n^{2}\times {\frac {n(n+1)}{2}}} \) matrix which, for any \( n\times n \) symmetric matrix A, transforms \( {\displaystyle vech(A)} \) into \( vec(A) \):
\( {\displaystyle D_{n}vech(A)=vec(A)}. \)
For the \( 2\times 2 \) symmetric matrix \( {\displaystyle A=\left[{\begin{smallmatrix}a&b\\b&d\end{smallmatrix}}\right]} \) , this transformation reads
\( {\displaystyle D_{n}vech(A)=vec(A)\implies {\begin{bmatrix}1&0&0\\0&1&0\\0&1&0\\0&0&1\end{bmatrix}}{\begin{bmatrix}a\\b\\d\end{bmatrix}} ={\begin{bmatrix}a\\b\\b\\d\end{bmatrix}}} \)
The explicit formula for calculating the duplication matrix for a \( n\times n \) matrix is:
\( {\displaystyle D_{n}^{T}=\sum \limits _{i\geq j}u_{ij}(vecT_{ij})^{T}} \)
Where:
\( u_{ij} \) is a unit vector of order \( {\displaystyle {\frac {1}{2}}n(n+1)} \) having the value 1 in the position \( {\displaystyle (j-1)n+i-{\frac {1}{2}}j(j-1)} \) and 0 elsewhere;
\( T_{ij} \) is a \( n\times n \) matrix with 1 in position \( {\displaystyle (i,j)} \) and \( {\displaystyle (j,i)} \) and zero elsewhere
Elimination matrix
An elimination matrix \( L_{n} \) is a \( {\displaystyle {\frac {n(n-1)}{2}}\times n^{2}} \) matrix which, for any \( n\times n \) matrix A, transforms \( vec(A) \) into \( {\displaystyle vech(A)} \) :
\({\displaystyle L_{n}vec(A)=vech(A)}. \) [1]
For the \( {\displaystyle 2\times 2} \) matrix \( {\displaystyle A=\left[{\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right]} \) , one choice for this transformation is given by
\( {\displaystyle L_{n}vec(A)=vech(A)\implies {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}a\\c\\b\\d\end{bmatrix}} ={\begin{bmatrix}a\\c\\d\end{bmatrix}}}. \)
Notes
Magnus & Neudecker (1980), Definition 3.1
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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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