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. The following properties hold:

${\begin{aligned}C_{1}(A)&=A\\[6pt]C_{n}(A)&=\det(A){\text{ if }}A{\text{ is }}n\times n\\[6pt]C_{k}(AB)&=C_{k}(A)C_{k}(B)\\[6pt]C_{k}(aX)&=a^{k}C_{k}(X)\\[6pt]{\text{For }}n\times n{\text{ identity }}I,C_{k}(I)&=I\,,{\text{ the }}\textstyle {{\binom {n}{k}}\times {\binom {n}{k}}}{\text{ identity }}\\[6pt]C_{k}(A^{T})&=C_{k}(A)^{T}\,,{\text{ over any field}}\\[6pt]C_{k}(A^{*})&=C_{k}(A)^{*}\,,{\text{ over }}\mathbb {C} \\[6pt]C_{k}(A^{-1})&=C_{k}(A)^{-1}\,,{\text{ for }}n\times n,{\text{ invertible }}A\end{aligned}}$

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.