# 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 $C_{r}(A)$.

1. 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. 2.

If $A$ is nonsingular, the $C_{r}(A)^{-1}=C_{r}(A^{-1})$.

3. 3.

If $A$ has complex entries, then $C_{r}(A^{*})=(C_{r}(A))^{*}$.

4. 4.

$C_{r}(A^{T})=(C_{r}(A))^{T}$

5. 5.

$C_{r}(\overline{A})=\overline{C_{r}(A)}$

6. 6.

For any $k\in F$ $C_{r}(kA)=k^{r}C_{r}(A)$

7. 7.

$C_{r}(I_{n})=I_{\binom{n}{r}}$

8. 8.

$\det(C_{r}(A))=\det(A)^{\binom{n-1}{r-1}}$ (Sylvester — Franke theorem)

Title compound matrix CompoundMatrix 2013-03-22 16:13:39 2013-03-22 16:13:39 Mathprof (13753) Mathprof (13753) 9 Mathprof (13753) Definition msc 15-00 rth adjugate Sylvester -Franke theorem