Alternant matrix

Alternant matrix

In linear algebra, an alternant matrix, is a matrix with a particular structure, in which successive columns have a particular function applied to their entries. An alternant determinant is the determinant of an alternant matrix. Such a matrix of size "m" × "n" matrix may be written out as

:M=egin{bmatrix}f_1(alpha_1) & f_2(alpha_1) & dots & f_n(alpha_1)\f_1(alpha_2) & f_2(alpha_2) & dots & f_n(alpha_2)\f_1(alpha_3) & f_2(alpha_3) & dots & f_n(alpha_3)\vdots & vdots & ddots &vdots \f_1(alpha_m) & f_2(alpha_m) & dots & f_n(alpha_m)\end{bmatrix}or more succinctly:M_{i,j} = f_j(alpha_i)for all indices "i" and "j". (Some authors use the transpose of the above matrix.)

Examples of alternant matrices include Vandermonde matrices, for which f_i(alpha)=alpha^{i-1} and Moore matrices for which f_i(alpha)=alpha^{q^{i-1.

Alternant matrices are used in coding theory in the construction of alternant codes.

ee also

* List of matrices

References

*
*


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Alternant code — In coding theory, alternant codes form a class of parameterised error correcting codes which generalise the BCH codes.DefinitionAn alternant code over GF( q ) of length n is defined by a parity check matrix H of alternant form H i , j = αji y i …   Wikipedia

  • alternant — 1. adjective Alternating from one row of a matrix to another. 2. noun a) An alternate form or allomorph. b) A determinant which is an alternating function …   Wiktionary

  • Vandermonde matrix — In linear algebra, a Vandermonde matrix, named after Alexandre Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row, i.e., an m × n matrix or …   Wikipedia

  • Moore matrix — In linear algebra, a Moore matrix, introduced by E. H. Moore (1896), is a matrix defined over a finite field. When it is a square matrix its determinant is called a Moore determinant (this is unrelated to the Moore determinant of a… …   Wikipedia

  • Альтернативная матрица — Не следует путать с Альтернирующей матрицей (англ.)русск.. Содержание …   Википедия

  • List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A …   Wikipedia

  • List of matrices — This page lists some important classes of matrices used in mathematics, science and engineering: Matrices in mathematics*(0,1) matrix a matrix with all elements either 0 or 1. Also called a binary matrix . *Adjugate matrix * Alternant matrix a… …   Wikipedia

  • Determinant — This article is about determinants in mathematics. For determinants in epidemiology, see Risk factor. In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific… …   Wikipedia

  • Hückel method — The Hückel method or Hückel molecular orbital method (HMO) proposed by Erich Hückel in 1930, is a very simple linear combination of atomic orbitals molecular orbitals (LCAO MO) method for the determination of energies of molecular orbitals of pi… …   Wikipedia

  • Oscillateur parametrique optique — Oscillateur paramétrique optique Un Oscillateur paramétrique optique (OPO) est une source de lumière cohérente et monochromatique. À partir d une onde laser de pompe de fréquence ωp, un OPO produit deux ondes de fréquences inférieures : le… …   Wikipédia en Français

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”