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. Compound matrices are closely related to exterior algebras.
Definition.
Let be an matrix with real or complex entries. If is a subset of and is a subset of , then the -submatrix of , written , is the submatrix formed from by retaining only those rows indexed by and those columns indexed by . If , then is the -minor of .
The "r"th compound matrix of is a matrix, denoted , is defined as follows. If , then is the unique matrix. Otherwise, has size formula_1. Its rows and columns are indexed by -element subsets of and , respectively, in their lexicographic order. The entry corresponding to subsets and is the minor .
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.
For example, consider the matrix
The rows are indexed by and the columns by . Therefore, the rows of are indexed by the sets
and the columns are indexed by
Using absolute value bars to denote determinants, the second compound matrix is
Properties.
Let be a scalar, be an matrix, and be an matrix. If is a positive integer, then denotes the identity matrix. The transpose of a matrix will be written , and the conjugate transpose by . Then:
Assume in addition that is a square matrix of size . Then:
Relation to exterior powers.
Give the standard coordinate basis . The th exterior power of is the vector space
whose basis consists of the formal symbols
where
Suppose that be an matrix. Then corresponds to a linear transformation
Taking the th exterior power of this linear transformation determines a linear transformation
The matrix corresponding to this linear transformation (with respect to the above bases of the exterior powers) is . Taking exterior powers is a functor, which means that
This corresponds to the formula . It is closely related to, and is a strengthening of, the Cauchy–Binet formula.
Relation to adjugate matrices.
Let be an matrix. Recall that its th higher adjugate matrix is the formula_14 matrix whose entry is
where, for any set of integers, is the sum of the elements of . The adjugate of is its 1st higher adjugate and is denoted . The generalized Laplace expansion formula implies
If is invertible, then
A concrete consequence of this is Jacobi's formula for the minors of an inverse matrix:
Adjugates can also be expressed in terms of compounds. Let denote the "sign matrix":
and let denote the "exchange matrix":
Then Jacobi's theorem states that the th higher adjugate matrix is:
It follows immediately from Jacobi's theorem that
Taking adjugates and compounds does not commute. However, compounds of adjugates can be expressed using adjugates of compounds, and vice versa. From the identities
and the Sylvester-Franke theorem, we deduce
The same technique leads to an additional identity,
Applications.
The computation of compound matrices appears in a wide array of problems.
Compound and adjugate matrices appear when computing determinants of linear combinations of matrices. It is elementary to check that, if and are matrices, then
It is also true that:
This has the immediate consequence
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.