Minimal polynomial: Difference between revisions
imported>Richard Pinch (section on Minimal polynomial of an algebraic number) |
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In [[linear algebra]] the '''minimal polynomial''' of a [[square matrix]] is the [[monic polynomial]] of least [[degree of a polynomial|degree]] which the matrix satisfies. | In [[linear algebra]] the '''minimal polynomial''' of a [[square matrix]] is the [[monic polynomial]] of least [[degree of a polynomial|degree]] which the matrix satisfies. | ||
Let ''A'' be an ''n''×''n'' matrix. The powers ''I''=''A''<sup>0</sup>,''A''<sup>1</sup>,...,''A''<sup>''n''²</sup> must be [[linear dependence|linearly dependent]] since the matrix ring has dimension ''n''<sup>2</sup>, and so ''A'' satisfies some polynomial. Hence it makes sense to define the minimal polynomial as the [[monic polynomial]] of least degree which ''A'' satisfies, or which annihilates ''A''. | Let ''A'' be an ''n''×''n'' matrix over a field ''F''. The powers ''I''=''A''<sup>0</sup>,''A''<sup>1</sup>,...,''A''<sup>''n''²</sup> must be [[linear dependence|linearly dependent]] since the matrix ring has dimension ''n''<sup>2</sup> as a [[vector space]] over ''F'', and so ''A'' satisfies some polynomial. Hence it makes sense to define the minimal polynomial as the [[monic polynomial]] of least degree which ''A'' satisfies, or which annihilates ''A''. | ||
A similar definition applies to the minimal polynomial of an [[endomorphism]] of a finite-dimensional [[vector space]]. | A similar definition applies to the minimal polynomial of an [[endomorphism]] of a finite-dimensional [[vector space]]. | ||
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==Minimal polynomial of an algebraic number== | ==Minimal polynomial of an algebraic number== | ||
The minimal poynomial of an [[algebraic number]] α is the rational [[polynomial]] of least [[degree of a polynomial|degree]] which has α as a root. | The minimal poynomial of an [[algebraic number]] α is the [[rational number|rational]] [[polynomial]] of least [[degree of a polynomial|degree]] which has α as a root. | ||
Revision as of 01:34, 22 December 2008
In linear algebra the minimal polynomial of a square matrix is the monic polynomial of least degree which the matrix satisfies.
Let A be an n×n matrix over a field F. The powers I=A0,A1,...,An² must be linearly dependent since the matrix ring has dimension n2 as a vector space over F, and so A satisfies some polynomial. Hence it makes sense to define the minimal polynomial as the monic polynomial of least degree which A satisfies, or which annihilates A.
A similar definition applies to the minimal polynomial of an endomorphism of a finite-dimensional vector space.
The polynomials which annihilate A form an ideal in the ring of polynomials, and this is a principal ideal domain: we deduce that the minimal polynomial actually divides all other polynomials which A satisfies.
Since A satisfies its own characteristic polynomial by the Cayley-Hamilton theorem, we deduce that the minimal polynomial divides the characteristic polynomial. However, the two polynomials have the same set of roots, namely the set of eigenvalues of A.
Minimal polynomial of an algebraic number
The minimal poynomial of an algebraic number α is the rational polynomial of least degree which has α as a root.