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<p>Deleted section on firefighting. Not relevant to the article and is incorrect in what it says anyway.</p> <p><b>New page</b></p><div>In [[mathematics]], especially in [[linear algebra]] and [[Matrix (mathematics)|matrix theory]], the &#039;&#039;&#039;duplication matrix&#039;&#039;&#039; and the &#039;&#039;&#039;elimination matrix&#039;&#039;&#039; 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.<br /> <br /> ==Duplication matrix==<br /> The duplication matrix &#039;&#039;D&#039;&#039;&lt;sub&gt;&#039;&#039;n&#039;&#039;&lt;/sup&gt; is the unique &#039;&#039;n&#039;&#039;&lt;sup&gt;2&lt;/sup&gt; × &#039;&#039;n&#039;&#039;(&#039;&#039;n&#039;&#039;+1)/2 matrix which, for any &#039;&#039;n&#039;&#039; × &#039;&#039;n&#039;&#039; [[symmetric matrix]] &#039;&#039;A&#039;&#039;, transforms vech(&#039;&#039;A&#039;&#039;) into vec(&#039;&#039;A&#039;&#039;):<br /> :&#039;&#039;D&#039;&#039;&lt;sub&gt;n&lt;/sub&gt; vech(&#039;&#039;A&#039;&#039;) = vec(&#039;&#039;A&#039;&#039;).<br /> <br /> For the 2×2 symmetric matrix &#039;&#039;A&#039;&#039; = &lt;math&gt;\left[\begin{smallmatrix} a &amp; b \\ b &amp; d \end{smallmatrix}\right]&lt;/math&gt;, this transformation reads<br /> : &lt;math&gt;\begin{bmatrix} 1&amp;0&amp;0 \\ 0&amp;1&amp;0 \\ 0&amp;1&amp;0 \\ 0&amp;0&amp;1 \end{bmatrix} \begin{bmatrix} a \\ b \\ d \end{bmatrix} = \begin{bmatrix} a \\ b \\ b \\ d \end{bmatrix}&lt;/math&gt;<br /> <br /> ==Elimination matrix==<br /> The elimination matrix &#039;&#039;L&#039;&#039;&lt;sub&gt;&#039;&#039;n&#039;&#039;&lt;/sup&gt; is the unique &#039;&#039;n&#039;&#039;(&#039;&#039;n&#039;&#039;+1)/2 × &#039;&#039;n&#039;&#039;&lt;sup&gt;2&lt;/sup&gt; matrix which, for any &#039;&#039;n&#039;&#039; × &#039;&#039;n&#039;&#039; matrix &#039;&#039;A&#039;&#039;, transforms vec(&#039;&#039;A&#039;&#039;) into vech(&#039;&#039;A&#039;&#039;):<br /> :&#039;&#039;L&#039;&#039;&lt;sub&gt;n&lt;/sub&gt; vec(&#039;&#039;A&#039;&#039;) = vech(&#039;&#039;A&#039;&#039;).&amp;nbsp;&lt;ref&gt;{{harvtxt|Magnus|Neudecker|1980}}, Definition 3.1&lt;/ref&gt;<br /> <br /> For the 2×2 matrix &#039;&#039;A&#039;&#039; = &lt;math&gt;\left[\begin{smallmatrix} a &amp; b \\ c &amp; d \end{smallmatrix}\right]&lt;/math&gt;, this transformation reads<br /> : &lt;math&gt;\begin{bmatrix} 1&amp;0&amp;0&amp;0 \\ 0&amp;1&amp;0&amp;0 \\ 0&amp;0&amp;0&amp;1 \end{bmatrix} \begin{bmatrix} a \\ c \\ b \\ d \end{bmatrix} = \begin{bmatrix} a \\ c \\ d \end{bmatrix}&lt;/math&gt;.<br /> <br /> ==Notes==<br /> &lt;references/&gt;<br /> <br /> ==References==<br /> *{{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}}.<br /> *Jan R. Magnus and Heinz Neudecker (1988), &#039;&#039;Matrix Differential Calculus with Applications in Statistics and Econometrics&#039;&#039;, Wiley. ISBN 0-471-98633-X.<br /> *Jan R. Magnus (1988), &#039;&#039;Linear Structures&#039;&#039;, Oxford University Press. ISBN 0-19-520655-X <br /> <br /> [[Category:Matrices]]<br /> <br /> [[de:Eliminationsmatrix]]</div>

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