OpenMath Content Dictionary: linalgeig1

Canonical URL:
http://www.openmath.org/cd/linalgeig1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
linalgeig1.ocd
CD as XML Encoded OpenMath:
linalgeig1.omcd
Defines:
eigenspace, eigenvalues, is_eigenvalue, is_eigenvector
Date:
2004-11-30
Version:
4 (Revision 1)
Review Date:
2006-03-30
Status:
experimental


     This document is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

     The copyright holder grants you permission to redistribute this 
     document freely as a verbatim copy. Furthermore, the copyright
     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
          document is stated prominently in the derived work.  Moreover if
          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
          is because the names `mathN' may be used by the OpenMath Society
          for future extensions.
       c) The derived work is distributed under terms that allow the
          compilation of derived works, but keep paragraphs a) and b)
          intact.  The simplest way to do this is to distribute the derived
          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

This CD defines symbols for basic linear algebra related to eigenvalues.

Regardless of the way of forming vectors and matrices, this CD deals with eigenvalues, eigenvectors and related concepts.


is_eigenvalue

Role:
application
Description:

This symbol represents a Boolean binary function, whose first argument should be a square matrix A over a ring R and whose second argument should be an element of the ring R. Here, the matrix A acts on (row) vectors from the right and the scalar lambda is written to the left of the vector v. When applied to A and lambda, it means that there is an eigenvector vector with eigenvalue lambda.

Commented Mathematical property (CMP):
If is_eigenvalue(A,lambda), there is a nonzero row vector v such that v * A = lambda * v.
Formal Mathematical property (FMP):
is_eigenvalue ( A , λ ) v . v 0 v A = λ v
Signatures:
sts


[Next: is_eigenvector] [Last: eigenspace] [Top]

is_eigenvector

Role:
application
Description:

This symbol represents a Boolean binary function, whose first argument should be a square matrix A over a ring R and whose second argument should be a vector v of size rowcount(A) over the ring R.

When applied to A and v, it means that v is a left eigenvector of A.

Commented Mathematical property (CMP):
If is_eigenvector(A,v) then there is a scalar lambda such that v * A = lambda * v
Formal Mathematical property (FMP):
is_eigenvector ( A , v ) λ . v A = λ v
Signatures:
sts


[Next: eigenvalues] [Previous: is_eigenvalue] [Top]

eigenvalues

Role:
application
Description:

This symbol represents a binary function, whose arguments should be a square matrix A and a field F over which the matrix A is defined. When applied to A and F, it returns a vector whose entries are the eigenvalues of A contained in F, with multiplicities.

Example:
Consider the matrix A given by [[0,1], [1,-1] ]. Its characteristic polynomial is X^2+X-1, and its eigenvalues are the roots of this equation. Hence, over the rationals, there are no eigenvalues:
eigenvalues ( 0 1 1 -1 , Q ) = ( )
Extending the field by solutions of the equation, X^2+x-1= 0, we find two eigenvalues:
eigenvalues ( 0 1 1 -1 , quotient_ring ( poly_ring ( Q , X ) , principal_ideal ( poly_ring ( Q , X ) , X 2 + X - 1 ) ) )
could represent
( expression ( , X ) , expression ( expression ( - ( X + 1 ) ) ) )
or the vector whose components are interchanged. (The order of the eigenvalues is not specified.)
Commented Mathematical property (CMP):
The size of the vector eigenvalues(A,F) is at most rowcount(A).
Formal Mathematical property (FMP):
size ( eigenvalues ( A , F ) ) rowcount ( A )
Signatures:
sts


[Next: eigenspace] [Previous: is_eigenvector] [Top]

eigenspace

Role:
application
Description:

This symbol represents a binary function, whose arguments should be a square matrix A over a field F and an element lambda of F. When applied to A and lambda, it returns a matrix whose rows are a basis of the eigenspace of A with respect to lambda.

Signatures:
sts


[First: is_eigenvalue] [Previous: eigenvalues] [Top]