Compound matrix explained

In linear algebra, a branch of mathematics, a (multiplicative) compound matrix is a matrix whose entries are all minors, of a given size, of another matrix.[1] [2] [3] [4] Compound matrices are closely related to exterior algebras,[5] 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.[6]

Definition

Let be an matrix with real or complex entries. If is a subset of size of

Notes and References

  1. DeAlba, Luz M. Determinants and Eigenvalues in Hogben, Leslie (ed) Handbook of Linear Algebra, 2nd edition, CRC Press, 2013,, p. 4-4
  2. Gantmacher, F. R., The Theory of Matrices, volume I, Chelsea Publishing Company, 1959, p. 20
  3. Horn, Roger A. and Johnson, Charles R., Matrix Analysis, 2nd edition, Cambridge University Press, 2013,, p. 21
  4. Muldowney. James S.. 1990. Compound matrices and ordinary differential equations. Rocky Mountain Journal of Mathematics. en. 20. 4. 857–872. 10.1216/rmjm/1181073047. 0035-7596. free.
  5. Boutin. D.L.. R.F. Gleeson. R.M. Williams. Wedge Theory / Compound Matrices: Properties and Applications.. Office of Naval Research. https://web.archive.org/web/20210116083905/https://apps.dtic.mil/sti/pdfs/ADA320264.pdf. live. January 16, 2021. 1996. NAWCADPAX–96-220-TR.
  6. Bar-Shalom . Eyal . Dalin . Omri . Margaliot . Michael . 2023-03-15 . Compound matrices in systems and control theory: a tutorial . Mathematics of Control, Signals, and Systems . 35 . 3 . 467–521 . en . 10.1007/s00498-023-00351-8 . 2204.00676 . 247939832 . 0932-4194.

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