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)

Title	compound matrix
Canonical name	CompoundMatrix
Date of creation	2013-03-22 16:13:39
Last modified on	2013-03-22 16:13:39
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	9
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 15-00
Defines	rth adjugate
Defines	Sylvester -Franke theorem