Linear Algebra, 5th Edition

7.3 The Minimal Polynomial

The Cayley-Hamilton theorem (Theorem 5.22 p. 315) tells us that for any linear operator T on an n-dimensional vector space, there is a polynomial f(t) of degree n such that $f(\text{T})={\text{T}}_0$, namely, the characteristic polynomial of T. Hence there is a polynomial of least degree with this property, and this degree is at most n. If g(t) is such a polynomial, we can divide g(t) by its leading coefficient to obtain another polynomial p(t) of the same degree with leading coefficient 1, that is, p(t) is a monic polynomial. (See Appendix E.)

Definition.

Let T be a linear operator on a finite-dimensional vector space. A polynomial p(t) is called a minimal polynomial of T if p(t) is a monic polynomial of least positive degree for which ...