Characteristic polynomial

From Citizendium
Jump to: navigation, search
This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

In linear algebra the characteristic polynomial of a square matrix is a polynomial which has the eigenvalues of the matrix as roots.

Let A be an n×n matrix. The characteristic polynomial of A is the determinant

where X is an indeterminate and In is an identity matrix.

The characteristic polynomial is unchanged under similarity, and hence be defined for an endomorphism of a vector space, independent of choice of basis.

Properties

  • The characteristic polynomial is monic of degree n;
  • The set of roots of the characteristic polynomial is equal to the set of eigenvalues of A.

Cayley-Hamilton theorem

The Cayley-Hamilton theorem states that a matrix satisfies its own characteristic polynomial.