characteristic polynomial

Characteristic Polynomial of a Matrix

Let A be a n×n matrix over some field k. The characteristic polynomialMathworldPlanetmathPlanetmath pA(x) of A in an indeterminate x is defined by the determinantMathworldPlanetmath:



  • The polynomialPlanetmathPlanetmath pA(x) is an nth-degree polynomial over k.

  • If A and B are similar matricesMathworldPlanetmath, then pA(x)=pB(x), because

    pA(x) = det(A-xI)=det(P-1BP-xI)
    = det(P-1BP-P-1xIP)=det(P-1)det(B-xI)det(P)
    = det(P)-1det(B-xI)det(P)=det(B-xI)=pB(x)

    for some invertible matrix P.

  • The characteristic equation of A is the equation pA(x)=0, and the solutions to which are the eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of A.

Characteristic Polynomial of a Linear Operator

Now, let T be a linear operator on a vector spaceMathworldPlanetmath V of dimensionPlanetmathPlanetmath n<. Let α and β be any two ordered bases for V. Then we may form the matrices [T]α and [T]β. The two matrix representations of T are similar matrices, related by a change of bases matrix. Therefore, by the second remark above, we define the characteristic polynomial of T, denoted by pT(x), in the indeterminate x, by


The characteristic equation of T is defined accordingly.

Title characteristic polynomial
Canonical name CharacteristicPolynomial
Date of creation 2013-03-22 12:17:47
Last modified on 2013-03-22 12:17:47
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 15A18
Related topic Equation
Defines characteristic equation