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.

{\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 stated above is equivalent to the functoriality of the exterior power.