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Definition of minimal polynomial of a linear transformation

Created over 8 years ago, updated 10 days ago

Definition: The minimal polynomial of a linear transformation (T: V o V) is the unique monic polynomial (m_T(x)) of least degree such that (m_T(T) = 0) (the zero transformation).

Properties:

  • The minimal polynomial divides every polynomial (p(x)) for which (p(T) = 0)
  • The minimal polynomial divides the characteristic polynomial
  • The minimal polynomial and characteristic polynomial have the same irreducible factors (possibly with different multiplicities)
  • The minimal polynomial is unique

Example: For (T) represented by (A = egin{pmatrix} 2 & 0 \ 0 & 2 \end{pmatrix}), the characteristic polynomial is ((\lambda-2)^2) but the minimal polynomial is ((\lambda-2)) since (A - 2I = 0).