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The form is named after German mathematician Ferdinand Georg Frobenius. Some authors use the term rational canonical form for a somewhat different form that is more properly called the primary rational canonical form. Instead of decomposing into a minimum number of cyclic subspaces, the primary form decomposes into a maximum number of cyclic ...
The rational canonical form does not have these drawbacks: it exists over any field, is truly unique, and it can be computed using only arithmetic operations in the field; A and B are similar if and only if they have the same rational canonical form. The rational canonical form is determined by the elementary divisors of A; these can be ...
A canonical form is a labeled graph Canon(G) that is isomorphic to G, such that every graph that is isomorphic to G has the same canonical form as G. Thus, from a solution to the graph canonization problem, one could also solve the problem of graph isomorphism : to test whether two graphs G and H are isomorphic, compute their canonical forms ...
the linear mapping : makes a cyclic []-module, having a basis of the form {,, …,}; or equivalently [] / (()) as []-modules. If the above hold, one says that A is non-derogatory . Not every square matrix is similar to a companion matrix, but every square matrix is similar to a block diagonal matrix made of companion matrices.
In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization.
A canonical form is a labeled graph Canon(G) that is isomorphic to G, such that every graph that is isomorphic to G has the same canonical form as G. Thus, from a solution to the graph canonization problem, one could also solve the problem of graph isomorphism : to test whether two graphs G and H are isomorphic, compute their canonical forms ...
The Jordan form is used to find a normal form of matrices up to conjugacy such that normal matrices make up an algebraic variety of a low fixed degree in the ambient matrix space. Sets of representatives of matrix conjugacy classes for Jordan normal form or rational canonical forms in general do not constitute linear or affine subspaces in the ...
Such a quadratic irrational may also be written in another form with a square-root of a square-free number (for example (+) /) as explained for quadratic irrationals. By considering the complete quotients of periodic continued fractions, Euler was able to prove that if x is a regular periodic continued fraction, then x is a quadratic irrational ...
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