The Marvelous Minimal Polynomial Sheldon Axler

Sheldon Axler, San Francisco State University Suppose V is a finite-dimensional vector space and T is a linear map from V to V. 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. This expository talk will discuss several applications of the minimal polynomial that should be better known. The minimal polynomial of T is easily computable, usually very quickly even when V has high dimension. T is invertible if and only if the constant term of the minimal polynomial of T is nonzero. The zeros of the minimal polynomial of T 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, when V is a real or complex inner product space, the minimal polynomial leads to an easy proof of the finite-dimensional spectral theorem.