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In [[linear algebra]], a branch of [[mathematics]], a ('''multiplicative''') '''compound matrix''' is a [[matrix (mathematics)|matrix]] whose entries are all [[minor (linear algebra)|minors]], of a given size, of another matrix.<ref>DeAlba, Luz M. ''Determinants and Eigenvalues'' in Hogben, Leslie (ed) ''Handbook of Linear Algebra'', 2nd edition, CRC Press, 2013, {{isbn|978-1-4665-0729-6}}, p. 4-4</ref><ref>Gantmacher, F. R., ''The Theory of Matrices'', volume I, Chelsea Publishing Company, 1959, {{isbn|978-0-8218-1376-8}}p. 20</ref><ref>Horn, Roger A. and Johnson, Charles R., ''Matrix Analysis'', 2nd edition, Cambridge University Press, 2013, {{isbn|978-0-521-54823-6}}, p. 21</ref><ref name=":0">{{Cite journal|last=Muldowney|first=James S.|date=1990|title=Compound matrices and ordinary differential equations|url=http://projecteuclid.org/euclid.rmjm/1181073047|journal=Rocky Mountain Journal of Mathematics|language=en|volume=20|issue=4|pages=857–872|doi=10.1216/rmjm/1181073047|issn=0035-7596|via=|doi-access=free}}</ref> Compound matrices are closely related to [[exterior algebra]]s.
== Definition ==
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Let {{math|''A''}} be an {{math|''m'' × ''n''}} matrix with [[real number|real]] or [[complex number|complex]] entries.{{efn|The definition, and the purely algebraic part of the theory, of compound matrices requires only that the matrix have entries in a [[commutative ring]]. In this case, the matrix corresponds to a [[module homomorphism|homomorphism]] of [[finitely generated module|finitely generated]] [[free module]]s.}} If {{math|''I''}} is a [[subset]] of size {{math|''r''}} of {{math|{1, ..., ''m''<nowiki>}</nowiki>}} and {{math|''J''}} is a subset of size {{math|''s''}} of {{math|{1, ..., ''n''<nowiki>}</nowiki>}}, then the '''{{math|(''I'', ''J'' )}}-submatrix of {{math|''A''}}''', written {{math|''A''<sub>''I'', ''J''</sub>}} , is the submatrix formed from {{math|''A''}} by retaining only those rows indexed by {{math|''I''}} and those columns indexed by {{math|''J''}}. If {{math|1=''r'' = ''s''}}, then {{math|det ''A''<sub>''I'', ''J''</sub>}} is the '''{{math|(''I'', ''J'' )}}-[[minor (linear algebra)|minor]]''' of {{math|''A''}}.
The '''''r'' th compound matrix''' of {{math|''A''}} is a matrix, denoted {{math|''C''<sub>''r'' </sub>(''A'')}}, is defined as follows. If {{math|''r'' > min(''m'', ''n'')}}, then {{math|''C''<sub>''r'' </sub>(''A'')}} is the unique {{math|0 × 0}} matrix. Otherwise, {{math|''C''<sub>''r'' </sub>(''A'')}} has size <math display="inline">\binom{m}{r} \!\times\! \binom{n}{r}</math>. Its rows and columns are indexed by {{math|''r''}}-element subsets of {{math|{1, ..., ''m''<nowiki>}</nowiki>}} and {{math|{1, ..., ''n''<nowiki>}</nowiki>}}, respectively, in their [[lexicographic order]]. The entry corresponding to subsets {{math|''I''}} and {{math|''J''}} is the minor {{math|det ''A''<sub>''I'', ''J''</sub>}}.
In some applications of compound matrices, the precise ordering of the rows and columns is unimportant. For this reason, some authors do not specify how the rows and columns are to be ordered.<ref>Kung, Rota, and Yan, p. 305.</ref>
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