Compound matrix

In mathematics, the kth 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}}

References

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

External links

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
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

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

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