Compound matrix

In mathematics, the kth compound matrix (sometimes referred to as the kth multiplicative compound matrix) $C_{k}(A)$, 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 $C_{k}(A^{-1})=C_{k}(A)^{-1}$

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.

External links

• 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
• To efficiently calculate compound matrices see: Christos Kravvaritis and Marilena Mitrouli, "Compound matrices: properties, numerical issues and analytical computations]", Numerical Algorithms, Vol. 50, (2009), 155–177. doi:10.1007/s11075-008-9222-7