Compound matrix

In mathematics, the kth compound matrix (sometimes referred to as the kth multiplicative compound matrix) $C_{k}(A)$, of an $m\times n$ matrix A is the ${\binom {m}{k}}\times {\binom {n}{k}}$ matrix formed from the determinants of all $k\times k$ submatrices of A, i.e., all $k\times k$ minors, arranged with the submatrix index sets in lexicographic order.

Properties

Let a be a complex number, A be a m × n complex matrix, B be a n × p complex matrix and I_n the identity matrix of order n × n.

The following properties hold:

• $C_{1}(A)=A$
• If m = n (that is, A is a square matrix), then $C_{n}(A)=\det(A)$
• $C_{k}(AB)=C_{k}(A)C_{k}(B)$
• $C_{k}(a\,A)=a^{k}C_{k}(A)$
• $C_{k}(I_{n})=I_{\binom {n}{k}}$
• $C_{k}(A^{*})=C_{k}(A)^{*}$

For square $A$, $n\times n$:

• If A is invertible, then $C_{k}(A^{-1})=C_{k}(A)^{-1}$

• $\det(C_{k}(A))=(\det A)^{\binom {n-1}{k-1}}$ (Sylvester-Franke Theorem)

As in , introduce the sign matrix $S=$ diagonal matrix with entries alternating $\pm 1$ with $S_{11}=1$. And the reversal matrix $J$ with 1's on the antidiagonal and zeros elsewhere.

• $C_{k}(A)^{-1}=\det(A)^{-1}J(C_{n-k}(SAS))^{T}J$ (see below)

Compound matrices and adjugates

[See for a classical discussion related to this section.]

Recall the adjugate matrix is the transpose of the matrix of cofactors, signed minors complementary to single entries. Then we can write

$\operatorname {adj} (A)=JC_{n-1}(SAS)^{T}J$ 1

with $T$ denoting transpose.

The basic property of the adjugate is the relation

$A\operatorname {adj} (A)=\det(A)I$,

hence $C_{k}(A)C_{k}(\operatorname {adj} (A))=\det(A)^{k}I$ while

$C_{k}(A)\operatorname {adj} (C_{k}(A))=\det(C_{k}(A))I$ 2

Comparing these and using the Sylvester-Franke theorem yields the identity

• $\operatorname {adj} (C_{k}(A))=\det(A)^{{\binom {n-1}{k-1}}-k}C_{k}(\operatorname {adj} (A))$

---

Jacobi's Theorem on the Adjugate

Jacobi's Theorem extends (1) to higher-order minors :

$C_{k}(\operatorname {adj} (A))=(\det A)^{k-1}J(C_{n-k}(SAS))^{T}J$

expressing minors of the adjugate in terms of complementary signed minors of the original matrix.

Substituting into the previous identity and going back to (2) yields

$C_{k}(A)\,J(C_{n-k}(SAS))^{T}J=\det(A)I$ and hence the formula for the inverse of the compound matrix given above.

Applications

The computation of compound matrices appears in a wide array of problems.

For instance, if $A$ is viewed as the matrix of an operator in a basis $(e_{1},\dots ,e_{n})$ then the compound matrix $C_{k}(A)$ is the matrix of the $k$-th exterior power $A^{\wedge k}$ in the basis $(e_{i_{1}}\wedge \dots \wedge e_{i_{k}})_{i_{1}<\dots <i_{k}}$. In this formulation, the multiplicativity property $C_{k}(AB)=C_{k}(A)C_{k}(B)$ is equivalent to the functoriality of the exterior power.

Compound matrices also appears in the determinant of the sum of two matrices, as the following identity is valid:

$\det(A+B)=\det \left({\begin{bmatrix}A&I_{n}\end{bmatrix}}{\begin{bmatrix}I_{n}\\B\end{bmatrix}}\right)=C_{n}({\begin{bmatrix}A&I_{n}\end{bmatrix}})C_{n}\left({\begin{bmatrix}I_{n}\\B\end{bmatrix}}\right)$

Numerical computation

In general, the computation of compound matrices is non effective due to its high complexity. Nonetheless, there is some efficient algorithms available for real matrices with special structures.

References

• Gantmacher, F. R. and Krein, M. G., Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, Revised Edition. American Mathematical Society, 2002. ISBN 978-0-8218-3171-7