Compound matrix

In mathematics, the kth 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.

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

See also

To efficiently calculate compound matrices see: "Compound matrices: properties, numerical issues and analytical computations" - Christos Kravvaritis · Marilena Mitrouli - DOI 10.1007/s11075-008-9222-7

References

1. R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1990, pp. 19-20

2. Gantmacher, F. R. and Krein, M. G. "Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems", Revised Edition, http://www.ams.org/bookstore?fn=20&arg1=diffequ&ikey=CHEL-345-H