Characteristic polynomial

From Citizendium, the Citizens' Compendium

Jump to: navigation, search


This article is a stub and thus not approved.
Main Article
Talk
Related Articles  [?]
Bibliography  [?]
External Links  [?]
 
http://en.citizendium.org/wiki?title=Characteristic_polynomial&printable=yes
This is a draft article, under development and not meant to be cited but you can help to improve it. These unapproved articles are 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

\chi_A(X) = \det(A - XI_n) ,\,

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.

Retrieved from "http://en.citizendium.org/wiki/Characteristic_polynomial"
Views
Personal tools