Frobenius-Normalform

The Frobenius normal form (after Ferdinand Georg Frobenius ) or rational canonical form of a square matrix with entries in any body is a transformed matrix ( with invertible matrix ), which has a special clear form. " Clearly " because each matrix can be transformed in just a matrix of this form and then let two matrices therefore exactly transform into each other if they have the same Frobenius normal form. If that's the case, we also say that two matrices are similar, because they represent the same linear transformation with respect to different bases. Therefore for every linear mapping of a finite-dimensional vector space into a base relation to which it is represented in Frobenius normal form. There may be several such bases, the transformation matrix is thus not uniquely determined.

The Frobenius normal form can be regarded as an alternative to a hand between Jordan normal form ( which in turn is a generalization of the diagonal form ), which must not be assumed that the characteristic polynomial splits into linear factors. On the other hand, characterizes the lemma of Frobenius matrices similar to each other by the elementary divisors of its characteristic matrices and returns the Frobenius normal form as a normal form of the vector space under the action of a polynomial ring.

Generalization of the diagonalization

If a matrix is diagonalizable, breaks its characteristic polynomial in a noisy linear factors with eigenvalues ​​. The corresponding eigenvectors form a basis of the vector space in which each basis vector is represented by a multiple of itself.

In a non- diagonalizable matrix enough eigenvectors are not available for a base, or the characteristic polynomial decomposes into irreducible factors not all have degrees 1. To determine the Frobenius normal form of a base of vectors is then analogous to the last paragraph sought to be made ​​, etc. of certain products of the irreducible factors to zero. It turns out that this is possible and we finally obtain a representation is in the divisor of, divides, etc. The factor here are the basic vectors, the subspace will be displayed in due from and to the respect of these basis vectors by the matrix

( the entries not specified in this so-called companion matrix for polynomial 0) is shown. The entire vector space decomposes in such invariant subspaces, and can be total by the block diagonal matrix

Represent. She is the Frobenius normal form of.

A disadvantage is that the Frobenius normal form of a diagonal matrix having non- diagonal form with eigenvalues ​​1 and 2, but

Is. Remedied by the Weierstrass normal form, in which the companion matrix is replaced in the block diagonal matrix by the Begleitmatrizen the potencies of different irreducible factors, ie by about

If with. A matrix is ​​diagonalizable if all these factors are linear and none occurs in the second or higher power; So their Weierstrass normal form is then a diagonal matrix.

Lemma of Frobenius

The set of all polynomials which are expressions of the form, with coefficients which forms a ring, the so-called polynomial ring. If a matrix is given, you can get a product of polynomial and vector defined by who is subject to the expected associative and distributive laws. One speaks of an operation of the polynomial on the vector space is determined by the vector space to a module.

After selecting a base, you can specify a module isomorphism. His domain is the factor module of modulo, the term referred to in angle brackets ( in an ad hoc notation chosen ) the product of the columns of the characteristic matrix. This isomorphism transmits the operation of the polynomial, means for, and is defined by

The characteristic matrix with entries in the polynomial can be obtained by the elementary divisor algorithm in a matrix

Be converted with invertible, where divisor of is, divider etc, and the polynomials leading coefficient 1 have. These polynomials are called the Invariantenteiler the characteristic matrix, the powers of the irreducible factors of the hot elementary divisors, and is the characteristic polynomial of, because ( the determinant of the characteristic matrix does not change when multiplied by the invertible and ). is the minimal polynomial of.

Because of the invertibility of the module and is now not only isomorphic ( namely ), but also isomorphic to. This factor module decomposes into a direct sum; see also the theorem on invariant factors in finitely generated modules over a principal ideal ring. The operation of the polynomial on the direct addend represented by the companion matrix, when a base is selected as in the previous section, and for the operation of or on the whole module resulting in a representation through the Frobenius normal form.

A further matrix given, making this to another module. An isomorphism must the operation of transfer, ie, which means that is transformed by the matrix of with respect to the chosen base. Similarity of matrices and is therefore equivalent to isomorphism of the associated modules and; and their decomposition into invariant factors discussed above has shown that this isomorphism if and only present when the characteristic matrices and have the same elementary divisors. This statement is known as Frobenius lemma.

As a further consequence of what is shown there is the Cayley - Hamilton: The operation of the characteristic polynomial makes all direct summands to zero, because all the dividers are from. That's why is so results in a matrix, used in its characteristic polynomial, the zero mapping.

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