Compound matrix

In mathematics, the kth compound matrix (sometimes referred to as the kth multiplicative compound matrix) $C_{k}(A)$ ,^[1] of an $m\times n$ matrix A is the ${\binom {m}{k}}\times {\binom {n}{k}}$ matrix formed from the determinants of all $k\times k$ submatrices of A, i.e., all $k\times k$ minors, arranged with the submatrix index sets in lexicographic order.

Properties

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

The following properties hold:

$C_{1}(A)=A$
If m = n (that is, A is a square matrix), then $C_{n}(A)=\det(A)$
$C_{k}(AB)=C_{k}(A)C_{k}(B)$
$C_{k}(a\,A)=a^{k}C_{k}(A)$
$C_{k}(I_{n})=I_{\binom {n}{k}}$
$C_{k}(A^{*})=C_{k}(A)^{*}$
If A is invertible, then $C_{k}(A^{-1})=C_{k}(A)^{-1}$

Applications

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

For instance, if $A$ is viewed as the matrix of an operator in a basis $(e_{1},\dots ,e_{n})$ then the compound matrix $C_{k}(A)$ is the matrix of the $k$ -th exterior power $A^{\wedge k}$ in the basis $(e_{i_{1}}\wedge \dots \wedge e_{i_{k}})_{i_{1}<\dots <i_{k}}$ . In this formulation, the multiplicativity property $C_{k}(AB)=C_{k}(A)C_{k}(B)$ 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]

$\det(A+B)=\det \left({\begin{bmatrix}A&I_{n}\end{bmatrix}}{\begin{bmatrix}I_{n}\\B\end{bmatrix}}\right)=C_{n}({\begin{bmatrix}A&I_{n}\end{bmatrix}})C_{n}\left({\begin{bmatrix}I_{n}\\B\end{bmatrix}}\right)$

Numerical computation

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]

References

^ R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1990, pp. 19–20. ISBN 9780521386326
^ 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.
^ Joseph P.S. Kung, Gian-Carlo Rota, and Catherine H. Yan, Combinatorics: the Rota way, Cambridge University Press, 2009, p. 306. ISBN 9780521883894
^ 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.
^ 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.

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

[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.

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