Characteristic polynomial

The characteristic polynomial (CP ) is a term from the mathematical subfield of linear algebra. This polynomial, which is defined for square matrices and endomorphisms of finite dimensional vector spaces, gives information about some properties of the matrix or linear map.

Equation, where the characteristic polynomial is set equal to zero, is sometimes referred to as the secular equation. Their solutions are the eigenvalues ​​of the matrix or the linear mapping.

Definition

The characteristic polynomial of a square matrix with entries of a body is defined by:

Here, denotes the -dimensional unit matrix and the determinant.

Is one -dimensional vector space and an endomorphism, then the characteristic polynomial is given by:

A representation of the matrix of the endomorphism.

The characteristic polynomial is a monic polynomial of degree n in the polynomial ring. The notation for the characteristic polynomial is very inconsistent, other variants, for example, or Bourbaki.

The definition of the characteristic polynomial as is also common. Is odd, it differs by the factor of the above definition, that is, the polynomial is then not normalized.

Associated with eigenvalues

The characteristic polynomial play an important role in the determination of the eigenvalues ​​of a matrix, for the eigenvalues ​​are exactly the zeros of the characteristic polynomial. Even if it always selects a base and thus a representation matrix for explicitly calculating the characteristic polynomial, the polynomial as the determinant does not depend upon that choice.

In order to show that the eigenvalues ​​are precisely the zeros of the characteristic polynomial, proceed as follows:

It should be and a matrix over. Then the following equivalences apply:

Formulas and algorithms

If we write the characteristic polynomial in the form

As is always the trace and the determinant of.

Especially for matrices the characteristic polynomial thus has the particularly simple form

For matrices results in the form:

Here, the matrix, which is obtained by deleting the - th row and is -th column ( a minor ).

The coefficients of, with the help of suitable methods, such as the algorithm of Leverrier - Faddejew or the algorithm of Samuelson - Berkowitz, also determine systematically.

Properties

  • The characteristic polynomials of two similar matrices are equal. However, the converse is not true in general.
  • The matrix and its transpose have the same characteristic polynomial.
  • By the theorem of Cayley - Hamilton is a matrix zero point of its characteristic polynomial: .
  • The minimal polynomial of a linear map divides the characteristic polynomial.
  • If a matrix and a matrix so true.

Proof:

Example

Wanted is the characteristic polynomial of the matrix

As defined above, calculated is as follows:

Thus are 1, -1 and 4, the zeros of the characteristic polynomial, and therefore also the eigenvalues ​​of the matrix.

Identity matrix Degree of a polynomial Cayley–Hamilton theorem Minimal polynomial (linear algebra)
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