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OpenMath Content Dictionary: linalgeig2

Canonical URL:
http://www.openmath.org/cd/linalgeig2.ocd
CD Base:
http://www.openmath.org/cd
CD File:
linalgeig2.ocd
CD as XML Encoded OpenMath:
linalgeig2.omcd
Defines:
eigenvalue, eigenvector
Date:
2004-11-30
Version:
4 (Revision 1)
Review Date:
2006-03-30
Status:
experimental

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This CD defines symbols for basic linear algebra over a field of characteristic zero related to eigenvalues.

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

eigenvalue

Role:
application
Description:

This symbol represents a binary function. The first argument should be a square matrix A defined over the field of complex numbers, the second should be an index i to specify the eigenvalue. When applied to A and i it represents the i-th eigenvalue of A (counted without multiplicities). The ordering imposed on the eigenvalues is first on the modulus of the value, and second on the argument of the value. A definition of eigenvalue is given in Elementary Linear Algebra, Stanley I. Grossman in Definition 1 of chapter 6, page 533.

Commented Mathematical property (CMP):
eigenvector(A,i) * A = eigenvalue(A,i)*eigenvector(A,i)
Formal Mathematical property (FMP):
eigenvector ⁡ ( A , i ) ⁢ A = eigenvalue ⁡ ( A , i ) ⁢ eigenvector ⁡ ( A , i )
Signatures:
sts

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eigenvector

Role:
application
Description:

This symbol represents a binary function. Its first argument should be a square matrix A defined over the complex numbers, the second should be an index i to specify which eigenvalue this eigenvector should be paired with, with the ordering specified in the definition of eigenvalue in this CD. When applied to A and i, it represents an eigenvector for A with respect to the i-th eigenvalue.

Commented Mathematical property (CMP):
A*eigenvector(A) = eigenvalue(A)*eigenvector(A)
Formal Mathematical property (FMP):
A ⁢ eigenvector ⁡ ( A , i ) = eigenvalue ⁡ ( A , i ) ⁢ eigenvector ⁡ ( A , i )
Signatures:
sts

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