Compound matrix: Difference between revisions

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In mathematics, the ''k''th '''compound matrix''' (sometimes referred to as the ''k''th '''''multiplicative'' compound matrix''') <math>C_k(A)</math>,<ref>R.A. Horn and C.R. Johnson, ''Matrix Analysis'', Cambridge University Press, 1990, pp. 19–20</ref> of an <math>m\times n</math> [[matrix (mathematics)|matrix]] ''A'' is the <math>\binom m k\times \binom n k</math> matrix formed from the [[determinant]]s of all <math> k\times k</math> submatrices of ''A'', i.e., all <math>k\times k</math> minors, arranged with the submatrix index sets in [[lexicographic order]].

The following properties hold:

: <math>

\begin{align}

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If <math>A</math> is viewed as the matrix of an operator in a basis <math>(e_1,\dots,e_n)</math> then the compound matrix <math>C_k(A)</math> is the matrix of the <math>k</math>-th exterior power <math>A^{\wedge k}</math> in the basis <math>(e_{i_1} \wedge \dots \wedge e_{i_k}, )_{i_1 < \dots < i_k)}</math>. In this formulation, the multiplicativity property stated<math>C_k(AB) above= C_k(A)C_k(B)</math> is equivalent to the functoriality of the exterior power. <ref>Joseph P.S. Kung, Gian-Carlo Rota, and Catherine H. Yan, ''Combinatorics: the Rota way'', Cambridge University Press, 2009, p. 306.</ref>