OpenMath

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OpenMath Content Dictionary: linalgpoly1

Canonical URL:
http://www.openmath.org/cd/linalgpoly1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
linalgpoly1.ocd
CD as XML Encoded OpenMath:
linalgpoly1.omcd
Defines:
characteristic_poly, minimum_poly, substitute
Date:
2004-11-30
Version:
4 (Revision 1)
Review Date:
2006-03-30
Status:
experimental

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This CD defines the following symbols for polynomials related to basic linear algebra over a field: the characteristic and the minimum polynomial, as well as a substitution of a square matrix in a polynomial.

characteristic_poly

Role:
application
Description:

This symbol represents a binary function. This first argument should be a square matrix A defined over a field F, the second argument a variable X. When applied to A and X, it represents the characteristic polynomial of A in the variable X over the field F. (The output should be semantically equivalent to an object obtained by the poly_ring_d_named constructor of the CD polyd1.)

Example:
The characteristic polynomial of the matrix [[0,1],[-1,-1]] is equal to X^2+X+1.
characteristic_poly ⁡ ( 0 1 -1 -1 , X ) = expression ⁡ ( poly_ring_d_named ⁡ ( Q , X ) , X 2 + X + 1 )
Commented Mathematical property (CMP):
The characteristic polynomial with variable X is the determinant of the matrix A - X identity(columcount(A)).
Formal Mathematical property (FMP):
characteristic_poly ⁡ ( A , X ) = det A - X ⁢ identity ⁡ ( rowcount ⁡ ( A ) )
Signatures:
sts

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minimum_poly

Role:
application
Description:

This symbol represents a binary function. This first argument should be a square matrix A defined over a field F, the second argument a variable X. When applied to A and X, it represents the minimum polynomial of A in the variable X over the field F. (The output should be semantically equivalent to an object obtained by the poly_ring_d_named constructor of the CD polyd1.)

Example:
The minimum polynomial of the matrix [[0,1,0,0], [-1,-1,0,0], [0,0,0,1], [0,0,-1,-1] ] is equal to X^2+X+1.
minimum_poly ⁡ ( 0 1 0 0 -1 -1 0 0 0 0 0 1 0 0 -1 -1 , X ) = expression ⁡ ( poly_ring_d_named ⁡ ( Q , X ) , X 2 + X + 1 )
Commented Mathematical property (CMP):
The minimum polynomial is the polynomial f of minimal degree such that f(A) = 0.
Formal Mathematical property (FMP):
f_poly = characteristic_poly ⁡ ( A , X ) ⇒ substitute ⁡ ( f , A ) = zero ⁡ ( rowcount ⁡ ( A ) , rowcount ⁡ ( A ) ) ∧ ∀   g . g ≠ 0 ∧ eq ⁡ ( substitute ⁡ ( g , A ) , zero ⁡ ( rowcount ⁡ ( A ) , rowcount ⁡ ( A ) ) ) ⇒ degree ⁡ ( g , f )
Signatures:
sts

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substitute

Role:
application
Description:

This symbol represents a binary function. This first argument should be a polynomial f in a single variable X, the second should be a square matrix A defined over a field F. When applied to f and A, it represents the matrix obtained by replacing X by A and the constant term by the corresponding scalar matrix.

Example:
The minimum polynomial of the matrix Substituting [[0,1], [-1,-1], ] in the polynomial X + 1 gives the matrix [[1,1], [-1,0], ]
substitute ⁡ ( expression ⁡ ( poly_ring_d_named ⁡ ( Q , X ) , X + 1 ) , 1 1 -1 0 )
Signatures:
sts

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