Frobenius matrix

A Frobenius matrix is a special kind of square matrix from the branch of mathematics known as numerical mathematics. A matrix is a Frobenius matrix if it has the following three properties:
* all entries on the main diagonal are ones
* the entries below the main diagonal of at most one column are arbitrary
* every other entry is zero

The following matrix is an example.: $A=egin{pmatrix} 1 & 0 & 0 & cdots & 0 \ 0 & 1 & 0 & cdots & 0 \ 0 & a_{32} & 1 & cdots & 0 \vdots & vdots & vdots & ddots & vdots \ 0 & a_{n2} & 0 & cdots & 1 end{pmatrix}$

Frobenius matrices are invertible. The inverse of a Frobenius matrix is again a Frobenius matrix. It is equal to the original matrix with changed signs outside the main diagonal. The inverse of the example above is therefore:: $A^{-1}=egin{pmatrix} 1 & 0 & 0 & cdots & 0 \ 0 & 1 & 0 & cdots & 0 \ 0 & -a_{32} & 1 & cdots & 0 \vdots & vdots & vdots & ddots & vdots \ 0 & -a_{n2} & 0 & cdots & 1 end{pmatrix}$

Frobenius matrices are named after Ferdinand Georg Frobenius. An alternative name for this class of matrices is Gauss transformation, after Carl Friedrich Gauss [Golub and Van Loan, p. 95.] . They are used in the process of Gaussian elimination to represent the Gaussian transformations.

If a matrix is multiplied from the left with a Frobenius matrix, a linear combination ofthe remaining rows is added to a particular row of the matrix. Multiplication with the inverse matrix subtracts the corresponding linear combination from the given row. This corresponds to one of the elementary operations of Gaussian elimination (besides the operation of transposing the rows and multiplying a row with a scalar multiple).

Notes

References

* Gene H. Golub and Charles F. Van Loan (1996). "Matrix Computations", third edition, Johns Hopkins University Press. ISBN 0-8018-5413-X (hardback), ISBN 0-8018-5414-8 (paperback).

Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

Frobenius-Matrix — Eine Frobeniusmatrix ist eine spezielle Matrix aus dem mathematischen Teilgebiet der Numerik. Eine Matrix ist eine Frobeniusmatrix, wenn sie die folgenden drei Eigenschaften aufweist: auf der Hauptdiagonale stehen nur Einsen in höchstens einer… … Deutsch Wikipedia
Frobenius-Normalform — Die Frobenius Normalform (nach Ferdinand Georg Frobenius) oder rationale Normalform einer quadratischen Matrix A mit Einträgen in einem beliebigen Körper K ist eine transformierte Matrix T − 1AT (mit invertierbarer Matrix T), die eine spezielle… … Deutsch Wikipedia
Frobenius normal form — In linear algebra, the Frobenius normal form, Turner binormal projective form or rational canonical form of a square matrix A is a canonical form for matrices that reflects the structure of the minimal polynomial of A and provides a means of… … Wikipedia
Matrix norm — In mathematics, a matrix norm is a natural extension of the notion of a vector norm to matrices. Contents 1 Definition 2 Induced norm 3 Entrywise norms 3.1 Frobenius norm … Wikipedia
Frobenius theorem (differential topology) — In mathematics, Frobenius theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first order homogeneous linear partial differential equations. In modern geometric terms … Wikipedia
Matrix multiplication — In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n by m matrix and B is an m by p matrix, the result AB of their multiplication is an n by p matrix defined only if… … Wikipedia
Matrix ring — In abstract algebra, a matrix ring is any collection of matrices forming a ring under matrix addition and matrix multiplication. The set of n×n matrices with entries from another ring is a matrix ring, as well as some subsets of infinite matrices … Wikipedia
Frobenius group — In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non trivial elementfixes more than one point and some non trivial element fixes a point. They are named after F. G. Frobenius. Structure The… … Wikipedia
Frobenius algebra — In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality… … Wikipedia
Frobenius theorem — There are several mathematical theorems named after Ferdinand Georg Frobenius. They include:* Frobenius theorem in differential geometry and topology for integrable subbundles; * Frobenius theorem in abstract algebra characterizing the finite… … Wikipedia

Academic Dictionaries and Encyclopedias

Frobenius matrix

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

Frobenius matrix

Look at other dictionaries:

Share the article and excerpts

Direct link