Minimal polynomial (linear algebra)

Under a minimal polynomial is generally understood as a polynomial of minimal degree that just fulfilled a property that is no longer fulfilled by factors of lower degree. More: In various areas of mathematics, the minimal polynomial is the minimum linear dependence between the powers of a matrix, or more generally an element of an algebra.

Definition

It had a body and a unitary algebra. Then the minimal polynomial of an element is the normalized polynomial of smallest degree that has as root.

The minimal polynomial can also be normalized generator of the core of the homomorphism

The Einsetzungshomomorphismus are described, the ring of polynomials with coefficients in is.

In a finite-dimensional algebra, each element has a unique minimal polynomial, in an infinite- need not be true that. There are called the elements that have a minimal polynomial, algebraic elements over the body; Elements for which it does not, transcendental elements.

Linear Algebra

The minimal a square array on a body is the smallest -degree polynomial with coefficients normalized in such that ( zero matrix ) is.

The following statements are equivalent for from:

• Is a zero of, that is,,
• Zero of the characteristic polynomial of,
• Is an eigenvalue of.

The multiplicity of a zero of determining the length of the longest main vector chain with eigenvalue, that is, the multiplicity is for example 4, then there exists a chain of four mutually linearly independent principal vectors ( levels 1 to 4) corresponding to the eigenvalue. If other major vector chains for the eigenvalue exist that are linearly independent of the chain of length 4, then they are in no case longer. Thus, the size of the largest Jordan block belonging to the Jordan normal form of between identical to the multiplicity of the minimal polynomial.

Under the geometric multiplicity of the eigenvalue of is understood, however, the number of linearly independent eigenvectors for this eigenvalue. In other words, the geometric multiplicity of an eigenvalue of the square matrix is the dimension of the solution space of.

More generally, one can investigate (even without commitment to a particular base ) to an endomorphism of a vector space the core of Einsetzungshomomorphismus from the definition, this leads also in infinite-dimensional vector spaces to a minimal polynomial, if this core is not the zero vector space. A simple example is the projection images that are idempotent by definition, that satisfy the relation. Thus, each projection has one of the polynomials or as a minimal polynomial.

Body theory

In the field theory the minimal polynomial is a term that occurs in a field extension.

Let be a field extension, which polynomial to the indeterminate and is algebraic, that is, there exists with. Then, a polynomial exists ( called the minimal ) with the properties

If one considers the extension field as a vector space over a particular item and as endomorphism on ( by the picture), one arrives at an algebraic element to the same minimal polynomial (in the sense of linear algebra ) as in the field theory.

Properties

• Minimal polynomials are irreducible over the body.
• Every polynomial with coefficients in the base body, which has an algebraic element as zero, is a ( polynomial ) multiple of the minimal polynomial of.
• The degree of the minimal polynomial of is equal to the degree of easy expansion.

See also: splitting, Cayley - Hamilton

Examples

• Consider the field extension with the imaginary unit: The minimal polynomial of is because it has a zero, is normalized, and every polynomial of lower degree would be linear and would have only one zero in.
• The polynomial is not a minimal polynomial of any element of any extension because it can be used as display and for any of its zeros is a polynomial of smallest degree.

Examples of minimal polynomials of an algebraic element

• Minimal polynomials over from using any complex square root is:   're already a zero of. This polynomial is irreducible over but when. If, then the minimal polynomial

• Minimal polynomials over of: It is. So zero is. But this polynomial is not irreducible, because it has the factorization. Obviously, no zero of. So zero of has to be. And this polynomial is irreducible (eg by reduction modulo 2)