Compound matrix

In linear algebra, a branch of mathematics, a (multiplicative) compound matrix is a matrix whose entries are all minors, of a given size, of another matrix.^[1][2][3][4] Compound matrices are closely related to exterior algebras,^[5] and their computation appears in a wide array of problems, such as in the analysis of nonlinear time-varying dynamical systems and generalizations of positive systems, cooperative systems and contracting systems.^[4][6]

Definition

Let A be an m × n matrix with real or complex entries.^[a] If I is a subset of size r of {1, ..., m} and J is a subset of size s of {1, ..., n}, then the (I, J )-submatrix of A, written A_{I, J} , is the submatrix formed from A by retaining only those rows indexed by I and those columns indexed by J. If r = s, then det A_{I, J} is the (I, J )-minor of A.

The r th compound matrix of A is a matrix, denoted C_r(A), is defined as follows. If r > min(m, n), then C_r(A) is the unique 0 × 0 matrix. Otherwise, C_r(A) has size ${\textstyle {\binom {m}{r}}\!\times \!{\binom {n}{r}}}$. Its rows and columns are indexed by r-element subsets of {1, ..., m} and {1, ..., n}, respectively, in their lexicographic order. The entry corresponding to subsets I and J is the minor det A_{I, J}.

In some applications of compound matrices, the precise ordering of the rows and columns is unimportant. For this reason, some authors do not specify how the rows and columns are to be ordered.^[7]

For example, consider the matrix

${\displaystyle A={\begin{pmatrix}1&2&3&4\\5&6&7&8\\9&10&11&12\end{pmatrix}}.}$

The rows are indexed by {1, 2, 3} and the columns by {1, 2, 3, 4}. Therefore, the rows of C₂(A) are indexed by the sets

${\displaystyle \{1,2\}<\{1,3\}<\{2,3\}}$

and the columns are indexed by

${\displaystyle \{1,2\}<\{1,3\}<\{1,4\}<\{2,3\}<\{2,4\}<\{3,4\}.}$

Using absolute value bars to denote determinants, the second compound matrix is

${\displaystyle {\begin{aligned}C_{2}(A)&={\begin{pmatrix}\left|{\begin{smallmatrix}1&2\\5&6\end{smallmatrix}}\right|&\left|{\begin{smallmatrix}1&3\\5&7\end{smallmatrix}}\right|&\left|{\begin{smallmatrix}1&4\\5&8\end{smallmatrix}}\right|&\left|{\begin{smallmatrix}2&3\\6&7\end{smallmatrix}}\right|&\left|{\begin{smallmatrix}2&4\\6&8\end{smallmatrix}}\right|&\left|{\begin{smallmatrix}3&4\\7&8\end{smallmatrix}}\right|\\\left|{\begin{smallmatrix}1&2\\9&10\end{smallmatrix}}\right|&\left|{\begin{smallmatrix}1&3\\9&11\end{smallmatrix}}\right|&\left|{\begin{smallmatrix}1&4\\9&12\end{smallmatrix}}\right|&\left|{\begin{smallmatrix}2&3\\10&11\end{smallmatrix}}\right|&\left|{\begin{smallmatrix}2&4\\10&12\end{smallmatrix}}\right|&\left|{\begin{smallmatrix}3&4\\11&12\end{smallmatrix}}\right|\\\left|{\begin{smallmatrix}5&6\\9&10\end{smallmatrix}}\right|&\left|{\begin{smallmatrix}5&7\\9&11\end{smallmatrix}}\right|&\left|{\begin{smallmatrix}5&8\\9&12\end{smallmatrix}}\right|&\left|{\begin{smallmatrix}6&7\\10&11\end{smallmatrix}}\right|&\left|{\begin{smallmatrix}6&8\\10&12\end{smallmatrix}}\right|&\left|{\begin{smallmatrix}7&8\\11&12\end{smallmatrix}}\right|\end{pmatrix}}\\&={\begin{pmatrix}-4&-8&-12&-4&-8&-4\\-8&-16&-24&-8&-16&-8\\-4&-8&-12&-4&-8&-4\end{pmatrix}}.\end{aligned}}}$

Properties

Let c be a scalar, A be an m × n matrix, and B be an n × p matrix. For k a positive integer, let I_k denote the k × k identity matrix. The transpose of a matrix M will be written M^T, and the conjugate transpose by M^*. Then:^[8]

• C₀(A) = I₁, a 1 × 1 identity matrix.
• C₁(A) = A.
• C_r(cA) = c^rC_r(A).
• If rk A = r, then rk C_r(A) = 1.
• If 1 ≤ r ≤ n, then ${\displaystyle C_{r}(I_{n})=I_{\binom {n}{r}}}$.
• If 1 ≤ r ≤ min(m, n), then C_r(A^T) = C_r(A)^T.
• If 1 ≤ r ≤ min(m, n), then C_r(A^*) = C_r(A)^*.
• C_r(AB) = C_r(A) C_r(B), which is closely related to Cauchy–Binet formula.

Assume in addition that A is a square matrix of size n. Then:^[9]

• C_n(A) = det A.
• If A has one of the following properties, then so does C_r(A):
  • Upper triangular,
  • Lower triangular,
  • Diagonal,
  • Orthogonal,
  • Unitary,
  • Symmetric,
  • Hermitian,
  • Skew-symmetric (when r is odd),
  • Skew-hermitian (when r is odd),
  • Positive definite,
  • Positive semi-definite,
  • Normal.
• If A is invertible, then so is C_r(A), and C_r(A⁻¹) = C_r(A)⁻¹.
• (Sylvester–Franke theorem) If 1 ≤ r ≤ n, then ${\displaystyle \det C_{r}(A)=(\det A)^{\binom {n-1}{r-1}}}$.^[10][11]

Relation to exterior powers

See also: Exterior algebra

Give Rⁿ the standard coordinate basis e₁, ..., e_n. The r th exterior power of Rⁿ is the vector space

${\displaystyle \wedge ^{r}\mathbf {R} ^{n}}$

whose basis consists of the formal symbols

${\displaystyle \mathbf {e} _{i_{1}}\wedge \dots \wedge \mathbf {e} _{i_{r}},}$

where

${\displaystyle i_{1}<\dots <i_{r}.}$

Suppose that A is an m × n matrix. Then A corresponds to a linear transformation

${\displaystyle A\colon \mathbf {R} ^{n}\to \mathbf {R} ^{m}.}$

Taking the r th exterior power of this linear transformation determines a linear transformation

${\displaystyle \wedge ^{r}A\colon \wedge ^{r}\mathbf {R} ^{n}\to \wedge ^{r}\mathbf {R} ^{m}.}$

The matrix corresponding to this linear transformation (with respect to the above bases of the exterior powers) is C_r(A). Taking exterior powers is a functor, which means that^[12]

${\displaystyle \wedge ^{r}(AB)=(\wedge ^{r}A)(\wedge ^{r}B).}$

This corresponds to the formula C_r(AB) = C_r(A)C_r(B). It is closely related to, and is a strengthening of, the Cauchy–Binet formula.

Relation to adjugate matrices

See also: Adjugate matrix

Let A be an n × n matrix. Recall that its r th higher adjugate matrix adj_r(A) is the ${\textstyle {\binom {n}{r}}\!\times \!{\binom {n}{r}}}$ matrix whose (I, J ) entry is

${\displaystyle (-1)^{\sigma (I)+\sigma (J)}\det A_{J^{c},I^{c}},}$

where, for any set K of integers, σ(K) is the sum of the elements of K. The adjugate of A is its 1st higher adjugate and is denoted adj(A). The generalized Laplace expansion formula implies

${\displaystyle C_{r}(A)\operatorname {adj} _{r}(A)=\operatorname {adj} _{r}(A)C_{r}(A)=(\det A)I_{\binom {n}{r}}.}$

If A is invertible, then

${\displaystyle \operatorname {adj} _{r}(A^{-1})=(\det A)^{-1}C_{r}(A).}$

A concrete consequence of this is Jacobi's formula for the minors of an inverse matrix:

${\displaystyle \det(A^{-1})_{J^{c},I^{c}}=(-1)^{\sigma (I)+\sigma (J)}{\frac {\det A_{I,J}}{\det A}}.}$

Adjugates can also be expressed in terms of compounds. Let S denote the sign matrix:

${\displaystyle S=\operatorname {diag} (1,-1,1,-1,\ldots ,(-1)^{n-1}),}$

and let J denote the exchange matrix:

${\displaystyle J={\begin{pmatrix}&&1\\&\cdots &\\1&&\end{pmatrix}}.}$

Then Jacobi's theorem states that the r th higher adjugate matrix is:^[13][14]

${\displaystyle \operatorname {adj} _{r}(A)=JC_{n-r}(SAS)^{T}J.}$

It follows immediately from Jacobi's theorem that

${\displaystyle C_{r}(A)\,J(C_{n-r}(SAS))^{T}J=(\det A)I_{\binom {n}{r}}.}$

Taking adjugates and compounds does not commute. However, compounds of adjugates can be expressed using adjugates of compounds, and vice versa. From the identities

${\displaystyle C_{r}(C_{s}(A))C_{r}(\operatorname {adj} _{s}(A))=(\det A)^{r}I,}$

${\displaystyle C_{r}(C_{s}(A))\operatorname {adj} _{r}(C_{s}(A))=(\det C_{s}(A))I,}$

and the Sylvester-Franke theorem, we deduce

${\displaystyle \operatorname {adj} _{r}(C_{s}(A))=(\det A)^{{\binom {n-1}{s-1}}-r}C_{r}(\operatorname {adj} _{s}(A)).}$

The same technique leads to an additional identity,

${\displaystyle \operatorname {adj} (C_{r}(A))=(\det A)^{{\binom {n-1}{r-1}}-r}C_{r}(\operatorname {adj} (A)).}$

Compound and adjugate matrices appear when computing determinants of linear combinations of matrices. It is elementary to check that if A and B are n × n matrices then

${\displaystyle \det(sA+tB)=C_{n}\!\left({\begin{bmatrix}sA&I_{n}\end{bmatrix}}\right)C_{n}\!\left({\begin{bmatrix}I_{n}\\tB\end{bmatrix}}\right).}$

It is also true that:^[15][16]

${\displaystyle \det(sA+tB)=\sum _{r=0}^{n}s^{r}t^{n-r}\operatorname {tr} (\operatorname {adj} _{r}(A)C_{r}(B)).}$

This has the immediate consequence

${\displaystyle \det(I+A)=\sum _{r=0}^{n}\operatorname {tr} \operatorname {adj} _{r}(A)=\sum _{r=0}^{n}\operatorname {tr} C_{r}(A).}$

Numerical computation

In general, the computation of compound matrices is non-effective due to its high complexity. Nonetheless, there are some efficient algorithms available for real matrices with special structure.^[17]

Notes

1. The definition, and the purely algebraic part of the theory, of compound matrices requires only that the matrix have entries in a commutative ring. In this case, the matrix corresponds to a homomorphism of finitely generated free modules.