Characteristic Polynomial

• If , the polynomial in the indeterminate is called the characteristic polynomial of

  Similarly for linear operators, if we have basis then is called the characteristic polynomial defined as

• (Friedberg 5.8) The characteristic polynomial of is a polynomial of degree with leading coefficient

  • (Friedberg 5.8.1) Let and be the characteristic polynomial of . Then
    • A scalar is an eigenvalue of if and only if
    • has at most distinct eigenvalues.
  • (Friedberg 5.8.2) The same results in (Friedberg 5.8.1) hold for linear operators as well.

• (Friedberg 5.9) Let be a linear operator on a vector space and be an eigenvalue of . A vector is an eigenvector of corresponding to if and only if and .

• (Friedberg e5.1.12a) Similar matrices have the same characteristic polynomial.

• (Friedberg e5.1.14) and have the same characteristic polynomial, and hence the same eigenvalues.

• (Friedberg e5.1.22) Let be a linear operator on over . If (see notation in Polynomial Ring) and is an eigenvector of corresponding to , then

• (Friedberg 5.11) The characteristic polynomial of any diagonalizable linear operator can be factored into linear factors. We say that the characteristic polynomial in this case splits.

• The algebraic multiplicity of an eigenvalue is the largest such that is a factor of the characteristic polynomial . We denote this as for linear transformation .

  • An eigenvalue is simple if

• (Friedberg 5.28) Cayley-Hamilton Theorem. Let be a linear operator on a finite dimensional vector space and let be the characteristic polynomial of . Then

  That is, satisfies its own characteristic polynomial.

  • Proof: This is equivalent to showing that . The result clearly holds if . If then consider the -cyclic subspace of generated by , denoted . On one hand, we have that

    This in turn implies the characteristic polynomial of is expressible as

    And substituting we get

    Also, divides . Therefore .

  • The result is also true for matrices. In such a case if and is its characteristic polynomial, then

  • A generalization of this concept is the Minimal Polynomial

Links §

• Friedberg, Insel and Spence