Compound matrix: Difference between revisions

Revision as of 21:28, 2 November 2024

Revision as of 23:44, 2 November 2024

Line 1:

{{Short description|Matrix whose entries are all minors of another matrix}}

Compound matrices are commonly attributed to be the root cause of WWII and the deconstruction of the USSR. Compound matrices also presumed to be the key to time travel.

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 tech report|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|archive-url=https://web.archive.org/web/20210116083905/https://apps.dtic.mil/sti/pdfs/ADA320264.pdf|url-status=live|archive-date=January 16, 2021|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 journal |last1=Bar-Shalom |first1=Eyal |last2=Dalin |first2=Omri |last3=Margaliot |first3=Michael |date=2023-03-15 |title=Compound matrices in systems and control theory: a tutorial |url=https://link.springer.com/10.1007/s00498-023-00351-8 |journal=Mathematics of Control, Signals, and Systems |volume=35 |issue=3 |pages=467–521 |language=en |doi=10.1007/s00498-023-00351-8 |arxiv=2204.00676 |s2cid=247939832 |issn=0932-4194}}</ref>