Frobenius Normal Form

Given a linear transformation the vector space becomes a module that is a module over the ring of polynomials with coefficients in the field the vector space determines the field which can be taken to be the maximal field containing the entries of a matrix for the polynomial acts on a vector by the rational canonical form corresponds to writing as.

Frobenius normal form. Jordan normal form calculator. For all scalars and for all matrices. The frobenius normal form m of a matrix a with entries in a field f can be obtained from it by means of a similarity transformation what this implies is that there is an invertible matrix p with entries in f for which p 1 ap m every matrix can be transformed by a similarity transformation to its frobenius normal form. Let a k t and let m be a cyclic torsion a module hence m is finite dimensional as a k vector space.

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. Frobenius normal form throughout this section k will be field. Die frobenius normalform oder rationale normalform einer quadratischen matrix a displaystyle a mit einträgen in einem beliebigen körper k displaystyle k ist eine transformierte matrix t 1 a t displaystyle t 1 at die eine spezielle übersichtliche form hat. A matrix norm is a norm on the vector space thus the matrix norm is a function.

Stack exchange network consists of 176 q a communities including stack overflow the largest most trusted online community for developers to learn share their knowledge and build their careers. The first and second natural canonical forms of a matrix see also normal form are block diagonal with companion matrices as blocks. In rational canonical form. The frobenius norm sometimes also called the euclidean norm a term unfortunately also used for the vector norm is matrix norm of an matrix defined as the square root of the sum of the absolute squares of its elements.

The vectors form a so called krylov sequence of vectors for that is a sequence of vectors such that the are independent and is a linear combination of. Make a note of one significant different with the jordan canonical form. In the domain near the regular singular point at here and for an equation in normal form actually the cut along some ray is introduced because the solutions are expected to have an essential singularity at. Both are also known as the frobenius normal form of.

übersichtlich deswegen weil sich jede matrix in genau eine matrix dieser form transformieren lässt und sich. K is not assumed to be algebraically closed. That must satisfy the following properties.