Let X be a finite set, and 2X the set of its subsets.
1. Let A = (AST) be a square matrix where S,T ∈ 2X, and its entries AST = 1 if S ⊆ T, and 0 otherwise.
2. Let B = (BST) be a square matrix where S,T ∈ 2X, S,T ≠ ∅, and its entries BST = 1 if the intersection S∩T contains an odd number of elements, and 0 otherwise.
Exercise: Find out how A−1 and B−1 are related to A and B.
Question: What other similar interesting matrices are known?
@Igor Germanovich,
are A, B such that each row (column) corresponds to one set S ∈ 2X ... in other words, A and B are 2|X|-by-2|X| matrices?
A is a matrix representing the classical poset (2x,⊆). A partial order being reflexive, antisymmetric and transitive, A has corresponding properties:
- ASS=1 for any S⊆X
- AST=1 implies ATS=0 (whenever S,T are not equal)
- if an entry AST=0, the entry at the same position [ST] of the matrix A2 is 0
A is anti-symmetric (symmetric with respect to the antidiagonal of A). If we arrange appropriately the sets (corresponding rows of A), A is upper triangular.
- Then A-1 is also upper triangular with a diagonal full of 1s.
- If AST=0, then it is so for the term at the same position in A-1. All other terms of A-1 are 1 or -1.
- A+A-1 is an interesting anti-symmetric upper triangular matrix.
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