In linear algebra, the Frobenius normal form or rational canonical form of a square matrix A with entries in a field F is a canonical form for matrices obtained by conjugation by invertible matrices over F. The form reflects a minimal decomposition of the vector space into subspaces that are cyclic for A (i.e., spanned by some vector and its repeated images under A). Since only one normal form can be reached from a given matrix (whence the "canonical"), a matrix B is similar to A if and only if it has the same rational canonical form as A. Since this form can be found without any operations that might change when extending the field F (whence the "rational"), notably without factoring polynomials, this shows that whether two matrices are similar does not change upon field extensions. 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 subspaces. It is also defined over F, but has somewhat different properties: finding the form requires factorization of polynomials, and as a consequence the primary rational canonical form may change when the same matrix is considered over an extension field of F. This article mainly deals with the form that does not require factorization, and explicitly mentions "primary" when the form using factorization is meant.
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linear algebra, the Frobeniusnormalform or rational canonical form of a square matrix A with entries in a field F is a canonical form for matrices obtained...
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical...
\|\mathbf {A} +\mathbf {B} \|\leq \|\mathbf {A} \|+\|\mathbf {B} \|} The Frobenius norm is analogous to the dot product of Euclidean vectors; multiply matrix...
polynomial Frobeniusnormalform Jordan normalform, up to a permutation of the Jordan blocks Index of nilpotence Elementary divisors, which form a complete...
there are bases on which it has a simple form, although not as simple as the diagonal form. The Frobeniusnormalform does not need of extending the field...
elements not in any conjugate of H form a normal subgroup called the Frobenius kernel K. (This is a theorem due to Frobenius (1901); there is still no proof...
harmonic oscillator. Another elementary application is obtaining the Frobeniusnormalform, i.e. the companion matrix of a polynomial. The commutative rings...
of the eigenvalues to form singular values. Among complex matrices, all unitary, Hermitian, and skew-Hermitian matrices are normal, with all eigenvalues...
matrix form for K [ T ] / p ( T ) {\displaystyle K[T]/p(T)} , this yields various canonical forms: invariant factors + companion matrix yields Frobenius normal...
algebra Jordan normalform – Form of a matrix indicating its eigenvalues and their algebraic multiplicities Frobeniusnormalform – Canonical form of matrices...
unique either, but its Frobenius norm is uniquely determined by A (just because the Frobenius norm of A is equal to the Frobenius norm of U = D + N). It...
except for the entries in a single column. Such a matrix is also called a Frobenius matrix, a Gauss matrix, or a Gauss transformation matrix. A block triangular...
algebraic theory of fields, a normal basis is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the...
called a Weierstrass equation, and said to be in Weierstrass form, or Weierstrass normalform. The definition of elliptic curve also requires that the curve...
essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely...
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation...
general, such a structure is called a Frobenius algebra. As the name implies, these were introduced by Frobenius in the nineteenth century. They have been...
(n, h) is (φh−1(n−1), h−1). Pairs (n, eH) form a normal subgroup isomorphic to N, while pairs (eN, h) form a subgroup isomorphic to H. The full group...
generators. The normalizer therefore has order p⋅(p − 1) and is known as a Frobenius group Fp(p−1) (especially for p = 5), and is the affine general linear...
the two primary ways of forming a smaller group from a larger one. Any normal subgroup has a corresponding quotient group, formed from the larger group...
position and zeros elsewhere, a basis (orthonormal with respect to the Frobenius inner product) can be described as follows: E j j for 1 ≤ j ≤ n ( n...
\colon x\mapsto x^{q}} (the rth power of the Frobenius automorphism). This allows one to define a Hermitian form on an Fq2 vector space V, as an Fq-bilinear...
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