Compound matrix: Difference between revisions

Revision as of 11:29, 28 October 2012

Revision as of 04:32, 2 November 2012

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{{unreferenced|date=August 2012}}

In mathematics, the ''k''th '''compound matrix''' <math>C_k(A)</math>,

In mathematics, the ''k''th '''compound matrix''' ''C''_''k''(''A'') of an ''m''&nbsp;&times;&nbsp;''n'' [[matrix (mathematics)|matrix]] ''A'' is the <math>\left(\binom m k - 1\right)\times\left(\binom n k - 1\right)</math> matrix formed from the [[determinant]]s of all ''k''&nbsp;&times;&nbsp;''k'' submatrices of ''A'' arranged with the submatrix index sets in [[lexicographic order]].

<ref>R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1990, pp. 19-20</ref>

In of mathematics,an the ''k''th '''compound matrix''' ''C''<submath>''k''m\times n</submath>(''A'') of an ''m''&nbsp;&times;&nbsp;''n'' [[matrix (mathematics)|matrix]] ''A'' is the <math>\left(\binom m k - 1\right)\times\left( \binom n k - 1\right)</math> matrix formed from the [[determinant]]s of all ''<math> k''&nbsp;&\times;&nbsp;'' k''</math> submatrices of ''A'', arrangedi.e., withall the<math>k\times submatrixk</math> index sets in [[lexicographic order]].minors,

arranged with the submatrix index sets in [[lexicographic order]].

: <math>

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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) & = akC_ka^kC_k(X) \\[6pt]

C_k(I) & = I \\[6pt]

C_k(A^HT) & = C_k(A)^HT\, , \text{ over any field} \\[6pt]

C_k(A^T*) & = C_k(A)^T*\, , \text{ over } \mathbb{C} \\[6pt]

C_k(A^{-1}) & = C_k(A)^{-1}

\end{align}

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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]

==References==

{{reflist}}

[[Category:Matrices]]