compound matrix

Suppose that A is an m×n matrix with entries from a field F and 1rmin(m,n). The rth compound matrix or rth of A is the (mr)×(nr) matrix whose entries are detA[α,β]), αQr,m and βQr,n, arranged in lexicographic order and we use submatrix notation. The notation for this matrix is Cr(A).

  1. 1.

    Cr(AB)=Cr(A)Cr(B) when r is less than or equal to the number of rows or columns of A and B

  2. 2.

    If A is nonsingular, the Cr(A)-1=Cr(A-1).

  3. 3.

    If A has complex entries, then Cr(A*)=(Cr(A))*.

  4. 4.


  5. 5.


  6. 6.

    For any kF Cr(kA)=krCr(A)

  7. 7.


  8. 8.

    det(Cr(A))=det(A)(n-1r-1) (Sylvester — Franke theoremMathworldPlanetmath)

Title compound matrix
Canonical name CompoundMatrix
Date of creation 2013-03-22 16:13:39
Last modified on 2013-03-22 16:13:39
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 9
Author Mathprof (13753)
Entry type Definition
Classification msc 15-00
Defines rth adjugate
Defines Sylvester -Franke theorem