Central cylindrical projection: Difference between revisions

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In mathematics, the ''k''th '''compound matrix''' <math>C_k(A)</math>,<ref>R.A. Horn and C.R. Johnson, ''Matrix Analysis'', Cambridge University Press, 1990, pp. 19–20</ref> of an <math>m\times n</math> [[matrix (mathematics)|matrix]] ''A'' is the <math>\binom m k\times \binom n k</math> matrix formed from the [[determinant]]s of all <math> k\times k</math> submatrices of ''A'', i.e., all <math>k\times k</math> minors, arranged with the submatrix index sets in [[lexicographic order]].
: <math>
\begin{align}
C_1(A) & = A \\[6pt]
C_n(A) & =  \det(A)\text{ if }A\text{ is }n\times n \\[6pt]
C_k(AB) & = C_k(A)C_k(B) \\[6pt]
C_k(aX) & = a^kC_k(X) \\[6pt]
\text{For } n\times n \text{ identity } I, C_k(I) & = I\,, \text{ the }\textstyle{\binom n k\times \binom n k} \text{ identity }\\[6pt]
C_k(A^T) & = C_k(A)^T\,, \text{ over any field} \\[6pt]
C_k(A^*) & = C_k(A)^*\,, \text{ over } \mathbb{C} \\[6pt]
C_k(A^{-1}) & = C_k(A)^{-1}\,, \text{ for } n\times n, \text{ invertible } A
\end{align}
</math>
 
==References==
{{reflist}}
 
==External links==
* Gantmacher, F. R. and Krein, M. G., ''Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems'', Revised Edition, http://www.ams.org/bookstore?fn=20&arg1=diffequ&ikey=CHEL-345-H
* To efficiently calculate compound matrices see: [http://users.uoa.gr/~mmitroul/mmitroulweb/numalg09.pdf "Compound matrices: properties, numerical issues and analytical computations" - Christos Kravvaritis · Marilena Mitrouli - DOI 10.1007/s11075-008-9222-7]
 
[[Category:Matrices]]

Latest revision as of 19:34, 16 April 2014

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