# Compound matrix

In mathematics, the kth compound matrix (sometimes referred to as the kth multiplicative compound matrix) ${\displaystyle C_{k}(A)}$,[1] of an ${\displaystyle m\times n}$ matrix A is the ${\displaystyle {\binom {m}{k}}\times {\binom {n}{k}}}$ matrix formed from the determinants of all ${\displaystyle k\times k}$ submatrices of A, i.e., all ${\displaystyle k\times k}$ 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 In the identity matrix of order n × n.

The following properties hold:

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

## Applications

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

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

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

## 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]

## Notes

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

• 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