compound matrix

Suppose that $A$ is an $m\\times n$ matrix with entries from a field $F$ and $1\\leq r\\leq\\min(m,n)$ . The $r^{th}$ compound matrix or $r^{th}$ of $A$ is the $\\binom{m}{r}\\times\\binom{n}{r}$ matrix whose entries are $\\det A[\\alpha,\\beta])$ , $\\alpha\\in Q_{r,m}$ and $\\beta\\in Q_{r,n}$ , arranged in lexicographic order and we use submatrix notation.The notation for this matrix is $C_{r}(A)$ .

Properties

1.
$C_{r}(AB)=C_{r}(A)C_{r}(B)$ when $r$ is less than or equal to the number of rows or columns of $A$ and $B$
2.
If $A$ is nonsingular, the $C_{r}(A)^{-1}=C_{r}(A^{-1})$ .
3.
If $A$ has complex entries, then $C_{r}(A^{*})=(C_{r}(A))^{*}$ .
4.
$C_{r}(A^{T})=(C_{r}(A))^{T}$
5.
$C_{r}(\\overline{A})=\\overline{C_{r}(A)}$
6.
For any $k\\in F$ $C_{r}(kA)=k^{r}C_{r}(A)$
7.
$C_{r}(I_{n})=I_{\\binom{n}{r}}$
8.
$\\det(C_{r}(A))=\\det(A)^{\\binom{n-1}{r-1}}$ (Sylvester — Franke theorem)

单词	CompoundMatrix
释义	compound matrix Suppose that $A$ is an $m\\times n$ matrix with entries from a field $F$ and $1\\leq r\\leq\\min(m,n)$ . The $r^{th}$ compound matrix or $r^{th}$ of $A$ is the $\\binom{m}{r}\\times\\binom{n}{r}$ matrix whose entries are $\\det A[\\alpha,\\beta])$ , $\\alpha\\in Q_{r,m}$ and $\\beta\\in Q_{r,n}$ , arranged in lexicographic order and we use submatrix notation.The notation for this matrix is $C_{r}(A)$ . Properties 1. $C_{r}(AB)=C_{r}(A)C_{r}(B)$ when $r$ is less than or equal to the number of rows or columns of $A$ and $B$ 2. If $A$ is nonsingular, the $C_{r}(A)^{-1}=C_{r}(A^{-1})$ . 3. If $A$ has complex entries, then $C_{r}(A^{})=(C_{r}(A))^{}$ . 4. $C_{r}(A^{T})=(C_{r}(A))^{T}$ 5. $C_{r}(\\overline{A})=\\overline{C_{r}(A)}$ 6. For any $k\\in F$ $C_{r}(kA)=k^{r}C_{r}(A)$ 7. $C_{r}(I_{n})=I_{\\binom{n}{r}}$ 8. $\\det(C_{r}(A))=\\det(A)^{\\binom{n-1}{r-1}}$ (Sylvester — Franke theorem)
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