In mathematics, the kth compound matrix (sometimes referred to as the kth multiplicative compound matrix) , of an matrix A is the matrix formed from the determinants of all submatrices of A, i.e., all minors, arranged with the submatrix index sets in lexicographic order.
Let a be a complex number, A be a m × n complex matrix, B be a n × p complex matrix and In the identity matrix of order n × n.
The following properties hold:
- If m = n (that is, A is a square matrix), then
- If A is invertible, then
The computation of compound matrices appears in a wide array of problems.
For instance, if is viewed as the matrix of an operator in a basis then the compound matrix is the matrix of the -th exterior power in the basis . In this formulation, the multiplicativity property 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:
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.
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