Compound matrix

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In mathematics, the kth compound matrix (sometimes referred to as the kth multiplicative compound matrix) ,[1] of an matrix A is the matrix formed from the determinants of all submatrices of A, i.e., all minors, arranged with the submatrix index sets in lexicographic order.

Properties[edit]

Let a be a complex number, A be a m × n complex matrix, B be a n × p complex matrix and In the identity matrix of order n × n.

The following properties hold:

  • If m = n (that is, A is a square matrix), then
  • If A is invertible, then

Applications[edit]

The computation of compound matrices appears in a wide array of problems.[2]

For instance, if is viewed as the matrix of an operator in a basis then the compound matrix is the matrix of the -th exterior power in the basis . In this formulation, the multiplicativity property is equivalent to the functoriality of the exterior power.[3]

Compound matrices also appears in the determinant of the sum of two matrices, as the following identity is valid:[4]

Numerical computation[edit]

In general, the computation of compound matrices is non effective due to its high complexity. Nonetheless, there is some efficient algorithms available for real matrices with special structures.[5]

Notes[edit]

  1. ^ R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1990, pp. 19–20. ISBN 9780521386326
  2. ^ D.L., Boutin; R.F. Gleeson; R.M. Williams (1996). Wedge Theory / Compound Matrices: Properties and Applications (Technical report). Office of Naval Research. NAWCADPAX–96-220-TR. 
  3. ^ Joseph P.S. Kung, Gian-Carlo Rota, and Catherine H. Yan, Combinatorics: the Rota way, Cambridge University Press, 2009, p. 306. ISBN 9780521883894
  4. ^ Prells, Uwe; Friswell, Michael I.; Garvey, Seamus D. (2003-02-08). "Use of geometric algebra: compound matrices and the determinant of the sum of two matrices". Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 459 (2030): 273–285. doi:10.1098/rspa.2002.1040. ISSN 1364-5021. 
  5. ^ Kravvaritis, Christos; Mitrouli, Marilena (2009-02-01). "Compound matrices: properties, numerical issues and analytical computations" (PDF). Numerical Algorithms. 50 (2): 155. doi:10.1007/s11075-008-9222-7. ISSN 1017-1398. 

References[edit]

  • Gantmacher, F. R. and Krein, M. G., Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, Revised Edition. American Mathematical Society, 2002. ISBN 978-0-8218-3171-7