Compound matrix: Difference between revisions

Revision as of 16:03, 28 August 2012

Revision as of 18:49, 28 August 2012

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In mathematics, the ''k''th '''compound matrix''' ''C''_''k''(''A'') of an ''n''&nbsp;&times;&nbsp;''n'' [[matrix (mathematics)|matrix]] ''A'' is the <math>\left(\binom m k - 1\right)\times\left(\binom m k - 1\right)</math> matrix formed from the [[determinant]]s of all ''k''&nbsp;&times;&nbsp;''k'' submatrices of ''A'' arranged with the submatrix index sets in [[lexicographic order]].

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The kth '''compound [[matrix (mathematics)|matrix]]''' of A[m#n] is the m!(k!(m-k)!)-1#n!(k!(n-k)!)-1 matrix formed from the determinants of all k#k submatrices of A arranged with the submatrix index sets in lexicographic order. Within this section, we denote this matrix by Ck(A).

C1(A) = A

Cn(A[n#n]) = det(A)

Ck(AB) = Ck(A)Ck(B)

Ck(aX) = akCk(X)

Ck(I) = I

Ck(AH) = Ck(A)H

Ck(AT) = Ck(A)T

Ck(A-1) = Ck(A)-1

: <math>

\begin{align}

C_1(A) & = A \\[6pt]

CnC_n(A[n#n]) & = \det(A) \\[6pt]

CkC_k(AB) & = CkC_k(A)CkC_k(B) \\[6pt]

C_k(aX) & = akC_k(X) \\[6pt]

C_k(I) & = I \\[6pt]

C_k(A^H) & = C_k(A)^H \\[6pt]

C_k(A^T) & = C_k(A)^T \\[6pt]

CkC_k(A^{-1}) & = CkC_k(A)^{-1}

\end{align}

</math>