Friction loss

2014-01-21T09:21:15Z

213.216.144.200: Deleted section on firefighting. Not relevant to the article and is incorrect in what it says anyway.

In [[mathematics]], especially in [[linear algebra]] and [[Matrix (mathematics)|matrix theory]], the '''duplication matrix''' and the '''elimination matrix''' are [[linear transformation]]s used for transforming [[vectorization (mathematics)#Half-vectorization|half-vectorization]]s of [[matrix (mathematics)|matrices]] into [[vectorization (mathematics)|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'' = <math>\left[\begin{smallmatrix} a & b \\ b & d \end{smallmatrix}\right]</math>, this transformation reads
: <math>\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}</math>

==Elimination matrix==
The elimination matrix ''L''''n'' is the unique ''n''(''n''+1)/2 × ''n''2 matrix which, for any ''n'' × ''n'' matrix ''A'', transforms vec(''A'') into vech(''A''):
:''L''n vec(''A'') = vech(''A''). <ref>{{harvtxt|Magnus|Neudecker|1980}}, Definition 3.1</ref>

For the 2×2 matrix ''A'' = <math>\left[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right]</math>, this transformation reads
: <math>\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}</math>.

==Notes==
<references/>

==References==
*{{Citation | last1=Magnus | first1=Jan R. | last2=Neudecker | first2=Heinz | title=The elimination matrix: some lemmas and applications | doi=10.1137/0601049 | year=1980 | journal=Society for Industrial and Applied Mathematics. Journal on Algebraic and Discrete Methods | issn=0196-5212 | volume=1 | issue=4 | pages=422–449}}.
*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

[[Category:Matrices]]

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Friction loss