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Duplication and elimination matrices
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Duplication and elimination matrices : ウィキペディア英語版
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 ''D''''n'' is the unique ''n''2 × ''n''(''n''+1)/2 matrix which, for any ''n'' × ''n'' symmetric matrix ''A'', transforms vech(''A'') into vec(''A''):
:''D''n vech(''A'') = vec(''A'').
For the 2×2 symmetric matrix ''A'' = \left(a & b \\ b & d \end\right ), this transformation reads
: \begin 1&0&0 \\ 0&1&0 \\ 0&1&0 \\ 0&0&1 \end \begin a \\ b \\ d \end = \begin a \\ b \\ b \\ d \end

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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