Compound matrix: Difference between revisions

Revision as of 02:49, 12 March 2019

Revision as of 21:39, 15 March 2019

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** Normal.

* If {{math|''A''}} is invertible, then so is {{math|''C''_''r''(''A'')}}, and {{math|1=''C''_''r''(''A''{{i sup|−1}}) = ''C''_''r''(''A''){{i sup|−1}}}}.

* (Sylvester–Franke theorem) If {{math|1 &le; ''r'' &le; ''n''}}, then <math>\det C_r(A) = (\det A)^{\binom{n-1}{r-1}}</math>.<ref name="Tornheim1952">{{cite journal|last1=Tornheim|first1=Leonard|title=The Sylvester–Franke Theorem|journal=The American Mathematical Monthly|volume=59|issue=6|year=1952|pages=389389–391|issn=00029890|doi=10.2307/2306811|jstor=2306811}}</ref><ref>[[Harley Flanders]] (1953) "A Note on the Sylvester-Franke Theorem", [[American Mathematical Monthly]] 60: 543–5, {{mr|id=0057835}}</ref>

==Relation to exterior powers==

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and let {{math|''J''}} denote the ''[[exchange matrix]]'':

:<math>J = \begin{pmatrix} & & 1 \\ & \cdots & \\ 1 & & \end{pmatrix}.</math>

Then '''Jacobi's theorem''' states that the {{math|''r''}}th higher adjugate matrix is:<ref name="NambiarSreevalsan2001">{{cite journal|last1=Nambiar|first1=K.K.|last2=Sreevalsan|first2=S.|title=Compound matrices and three celebrated theorems|journal=Mathematical and Computer Modelling|volume=34|issue=3-43–4|year=2001|pages=251–255|issn=08957177|doi=10.1016/S0895-7177(01)00058-9}}</ref><ref name="Price1947">{{cite journal|last1=Price|first1=G. B.|authorlink=G. B. Price|title=Some Identities in the Theory of Determinants|journal=The American Mathematical Monthly|volume=54|issue=2|year=1947|pages=7575–90|issn=00029890|doi=10.2307/2304856|jstor=2304856}}</ref>

:<math>\operatorname{adj}_r(A) = JC_{n-r}(SAS)^TJ.</math>