Is it possible to represent a bijection using matrices? For example, if I have a bijection from a set of vectors back onto itself. A vector in the set could be v = [10 2 11], and the mapping would be 10 -> 2, 2 -> 11, and 11 -> 10 so that v' = [ 11 2 10]. Another vector in the set, y = [2 11 10] would be mapped to y' = [11 10 2]. Does there exist a matrix such that v' = Av and y' = Ay for all permutations of the vectors in the set?
2026-04-11 20:11:25.1775938285
Representing a bijection as a matrix
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Yes. They're called permutation matrices.
More generally, the collection of bijective maps among a finite set (i.e., "permutations") forms a group. The study of representing groups with matrices is called representation theory.