Content deleted Content added
m →top: Various citation & identifier cleanup, plus AWB genfixes (arxiv version pointless when published) |
Citation bot (talk | contribs) Alter: template type. Add: s2cid, eprint, class, authors 1-1. Removed parameters. Some additions/deletions were parameter name changes. | Use this bot. Report bugs. | Suggested by Headbomb | Linked from Wikipedia:WikiProject_Academic_Journals/Journals_cited_by_Wikipedia/Sandbox | #UCB_webform_linked 8/35 |
||
Line 1:
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,<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=https://apps.dtic.mil/sti/pdfs/ADA320264.pdf|year=1996|number=NAWCADPAX–96-220-TR}}</ref> and their computation appears in a wide array of problems, such as in the analysis of nonlinear time-varying dynamical systems and generalizations of positive systems, cooperative systems and contracting systems.<ref name=":0" /><ref>{{cite
== Definition ==
Line 128:
Compound and adjugate matrices appear when computing determinants of [[linear combination]]s of matrices. It is elementary to check that if {{math|''A''}} and {{math|''B''}} are {{math|''n'' × ''n''}} matrices then
:<math>\det(sA + tB) = C_n\!\left(\begin{bmatrix} sA & I_n \end{bmatrix}\right)C_n\!\left(\begin{bmatrix} I_n \\ tB \end{bmatrix}\right).</math>
It is also true that:<ref>{{Cite journal|
:<math>\det(sA + tB) = \sum_{r=0}^n s^r t^{n-r} \operatorname{tr}(\operatorname{adj}_r(A)C_r(B)).</math>
This has the immediate consequence
Line 134:
== Numerical computation ==
In general, the computation of compound matrices is non-effective due to its high complexity. Nonetheless, there are some efficient algorithms available for real matrices with special structure.<ref>{{Cite journal|
==Notes==
| |||