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Matrix congruence is an equivalence relation. Matrix congruence arises when considering the effect of change of basis on the Gram matrix attached to a bilinear form or quadratic form on a finite-dimensional vector space : two matrices are congruent if and only if they represent the same bilinear form with respect to different bases .
It is easy to show that the trace of a matrix representing an element of Γ(N) cannot be −1, 0, or 1, so these subgroups are torsion-free groups. (There are other torsion-free subgroups.) The principal congruence subgroup of level 2, Γ(2), is also called the modular group Λ. Since PSL(2, Z/2Z) is isomorphic to S 3, Λ is a subgroup of index 6.
Integers in the same congruence class a ≡ b (mod n) satisfy gcd(a, n) = gcd(b, n); hence one is coprime to n if and only if the other is. Thus the notion of congruence classes modulo n that are coprime to n is well-defined. Since gcd(a, n) = 1 and gcd(b, n) = 1 implies gcd(ab, n) = 1, the set of classes coprime to n is closed under ...
Hill's cipher machine, from figure 4 of the patent. In classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra.Invented by Lester S. Hill in 1929, it was the first polygraphic cipher in which it was practical (though barely) to operate on more than three symbols at once.
If is the group of invertible real matrices, and is the subgroup of real matrices with determinant 1, then is normal in (since it is the kernel of the determinant homomorphism). The cosets of N {\displaystyle N} are the sets of matrices with a given determinant, and hence G / N {\displaystyle G\,/\,N} is isomorphic to the multiplicative group ...
A congruence on an algebra is an equivalence relation that forms a subalgebra of considered as an algebra with componentwise operations. One can make the set of equivalence classes A / Φ {\displaystyle A/\Phi } into an algebra of the same type by defining the operations via representatives; this will be well-defined since Φ {\displaystyle ...
The congruence relation, modulo m, partitions the set of integers into m congruence classes. Operations of addition and multiplication can be defined on these m objects in the following way: To either add or multiply two congruence classes, first pick a representative (in any way) from each class, then perform the usual operation for integers on the two representatives and finally take the ...
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. [1]