Lecture 26 - More on Minimal & Monic Polynomials

We're given some $T \in L (V)$ where $V$ is a complex, finite dimensional vector space, with distinct eigenvalues $λ_{1}, . . ., λ_{m}$ with multiplicities $d_{1}, . . ., d_{m}$ . Consider $p_{T}$ , the characteristic polynomial of transformation $T$ :

p_{T} (z) = \prod_{i = 1}^{m} (z - λ_{i})^{d_{i}}

Notice then $n = \dim (V) = \sum_{i = 1}^{m} d_{i}$ showing $\deg (p_{T)} = n$ . Furthermore, the zeroes of $p_{T}$ are precisely the eigenvalues of $T$ .

But we also found instances where we could reduce some of the $d_{i}$ 's to make a minimal polynomial. Specifically, the Cayley-Hamilton theorem states that $p_{T} (T) = \vec{0}$ . So every operator is a zero of its own characteristic polynomial. But we can reduce the degree of $p_{T}$ to see if this is still true for removed multiplicities.

Minimal Polynomial

Suppose $T \in L (V)$ . Then there is a unique monic polynomial $p$ of smallest degree such that $p (T) = \vec{0}$ .

Axler doesn't use notation for the minimal polynomial, so we'll use $μ_{T}$ to indicate the minimal polynomial. See the proof from the book for more details on this.