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{align} 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^kC_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{align}

References

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

External links