Compound matrix: Difference between revisions

Revision as of 23:28, 19 January 2018

Revision as of 00:18, 20 January 2018

Line 12:

* <math>C_k(A^{*}) = C_k(A)^{*}</math>

* If ''A'' is invertible, then <math>C_k(A^{-1}) = C_k(A)^{-1}</math>

If <math>A</math> is viewed as the matrix of an operator in a [[Basis (linear algebra)|basis]] <math>(e_1,\dots,e_n)</math> then the compound matrix <math>C_k(A)</math> is the matrix of the <math>k</math>-th [[Exterior algebra|exterior power]] <math>A^{\wedge k}</math> in the basis <math>(e_{i_1} \wedge \dots \wedge e_{i_k})_{i_1 < \dots < i_k}</math>. In this formulation, the multiplicativity property <math>C_k(AB) = C_k(A)C_k(B)</math> is equivalent to the [[Exterior algebra#Functoriality|functoriality]] of the exterior power.<ref>Joseph P.S. Kung, Gian-Carlo Rota, and [[Catherine Yan|Catherine H. Yan]], ''Combinatorics: the Rota way'', Cambridge University Press, 2009, p. 306. {{isbn|9780521883894}}</ref>

== Applications ==

The computation of compound matrices appears in a wide array of problems.<ref>{{cite techreport|first=Boutin|last=D.L.|author2=R.F. Gleeson|author3=R.M. Williams|title=Wedge Theory / Compound Matrices: Properties and Applications.|institution=Office of Naval Research|url=http://handle.dtic.mil/100.2/ADA320264|year=1996|number=NAWCADPAX–96-220-TR}}</ref>

IfFor instance, if <math>A</math> is viewed as the matrix of an operator in a [[Basis (linear algebra)|basis]] <math>(e_1,\dots,e_n)</math> then the compound matrix <math>C_k(A)</math> is the matrix of the <math>k</math>-th [[Exterior algebra|exterior power]] <math>A^{\wedge k}</math> in the basis <math>(e_{i_1} \wedge \dots \wedge e_{i_k})_{i_1 < \dots < i_k}</math>. In this formulation, the multiplicativity property <math>C_k(AB) = C_k(A)C_k(B)</math> is equivalent to the [[Exterior algebra#Functoriality|functoriality]] of the exterior power.<ref>Joseph P.S. Kung, Gian-Carlo Rota, and [[Catherine Yan|Catherine H. Yan]], ''Combinatorics: the Rota way'', Cambridge University Press, 2009, p. 306. {{isbn|9780521883894}}</ref>

Compound matrices also appears in the determinant of the sum of two matrices, as the following identity is valid:<ref>{{Cite journal|last=Prells|first=Uwe|last2=Friswell|first2=Michael I.|last3=Garvey|first3=Seamus D.|date=2003-02-08|title=Use of geometric algebra: compound matrices and the determinant of the sum of two matrices|url=http://rspa.royalsocietypublishing.org/content/459/2030/273|journal=Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences|language=en|volume=459|issue=2030|pages=273–285|doi=10.1098/rspa.2002.1040|issn=1364-5021}}</ref><blockquote><math>\det(A+B)=\det\left(\begin{bmatrix} A & I_n \end{bmatrix} \begin{bmatrix} I_n \\ B \end{bmatrix}\right)

= C_n(\begin{bmatrix} A & I_n \end{bmatrix}) C_n\left( \begin{bmatrix} I_n \\ B \end{bmatrix} \right)</math></blockquote>

== Numerical computation ==

In general, the computation of compound matrices is non effective due to its high complexity. Nonetheless, there is some efficient algorithms available for real matrices with special structures.<ref>{{Cite journal|last=Kravvaritis|first=Christos|last2=Mitrouli|first2=Marilena|date=2009-02-01|title=Compound matrices: properties, numerical issues and analytical computations|url=http://users.uoa.gr/~mmitroul/mmitroulweb/numalg09.pdf|journal=Numerical Algorithms|language=en|volume=50|issue=2|pages=155|doi=10.1007/s11075-008-9222-7|issn=1017-1398}}</ref>

==References==

{{reflist}}

==External links==

* Gantmacher, F. R. and Krein, M. G., ''Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems'', Revised Edition. American Mathematical Society, 2002. {{isbn|978-0-8218-3171-7}}

* To efficiently calculate compound matrices see: Christos Kravvaritis and Marilena Mitrouli, "[http://users.uoa.gr/~mmitroul/mmitroulweb/numalg09.pdf Compound matrices: properties, numerical issues and analytical computations]]", ''[[Numerical Algorithms]]'', Vol. 50, (2009), 155–177. {{doi|10.1007/s11075-008-9222-7}}

[[Category:Matrices]]