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The converse (inverse) of a transitive relation is always transitive. For instance, knowing that "is a subset of" is transitive and "is a superset of" is its converse, one can conclude that the latter is transitive as well. The intersection of two transitive relations is always transitive. [4]
The property of two matrices commuting is not transitive: A matrix may commute with both and , and still and do not commute with each other. As an example, the identity matrix commutes with all matrices, which between them do not all commute. If the set of matrices considered is restricted to Hermitian matrices without multiple eigenvalues ...
In the monoid of binary endorelations on a set (with the binary operation on relations being the composition of relations), the converse relation does not satisfy the definition of an inverse from group theory, that is, if is an arbitrary relation on , then does not equal the identity relation on in general.
In linear algebra, an invertible matrix is a square matrix which has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. An invertible matrix multiplied by its inverse yields the identity matrix. Invertible matrices are the same size as their ...
Matrix entries are given by the divisor function; entires of the inverse are given by the Möbius function. a ij are 1 if i divides j or if j = 1; otherwise, a ij = 0. A (0, 1)-matrix. Shift matrix: A matrix with ones on the superdiagonal or subdiagonal and zeroes elsewhere. a ij = δ i+1,j or a ij = δ i−1,j
A corollary of this result of Galois is that, if p is an odd prime number, then the order of a solvable transitive group of degree p is a divisor of (). In fact, every transitive group of prime degree is primitive (since the number of elements of a partition fixed by G must be a divisor of p ), and p ( p − 1 ) {\displaystyle p(p-1)} is the ...
The inverse of an integer matrix is again an integer matrix if and only if the determinant of equals or . Integer matrices of determinant 1 {\displaystyle 1} form the group S L n ( Z ) {\displaystyle \mathrm {SL} _{n}(\mathbf {Z} )} , which has far-reaching applications in arithmetic and geometry .
Similarly, the reflexive transitive symmetric closure or equivalence closure of a relation is the smallest equivalence relation that contains it. Other examples [ edit ]