The Marvelous Minimal Polynomial
Sheldon Axler
For each operator T on a finite-dimensional vector space, there is a unique monic polynomial p of smallest degree such that p(T) = 0 (here monic means that the coefficient of the highest order term equals 1). This polynomial p is called the minimal polynomial of T. The minimal polynomial should make a more important appearance in upper-division linear algebra courses than is usually the case.
As will be discussed in this talk, the minimal polynomial is easily computable, and its zeros are exactly the eigenvalues of T. Furthermore, the minimal polynomial of T tells us whether or not T is diagonalizable. The minimal polynomial of T also tells us whether or not there exists a basis of V with respect to which T has an upper-triangular matrix. In addition, we will see that consideration of the minimal polynomial leads to an easy proof of the finite-dimensional spectral theorem.
