If I have the minimal polynomial for $T$, is the minimal polynomial for$ T^{-1}$ the inverse of that polynomial?

Question

So I was given the minimal polynomial $p(z) = 5-3z+8z^2+2z^3-z^4$ for $T$. I'm trying to find the minimal polynomial for $T^{-1}$ and was wondering if the inverse function of $p$ is same as the minimal polynomial of $T^{-1}$. Any help is appreciated.

Answer 1

"Inverse", but in a different sense. Note that you are given $$ 5v-3Tv+8T^2v+2T^3v-T^4=0$$ for all $v$. Now apply $T^{-4}$ to this and obtain $$ 5T^{-4}v-3T^{-3}v+8T^{-2}v+2T^{-1}v-v=0$$ for all $v$, so the minimal poylnomial is $\frac15(5z^4-3z^3+8z^2+2z-1)$, in other words, i.e. formally $$q(x)=\frac{x^{\deg p}p(1/x)}{p(0)}$$ We neither have $q(p(x))=p(q(x))$ for all $x$ nor $p(x)q(x)=1$ for all $x$ nor $q(x)+p(x)=0$ for all $x$, so $q$ is not the inverese of $p$ in either of several ways to interprete the notion (compositional, multiplicative, additive).

Answer 2

I just want to add to @Hagen 's answer that there is another expression for the minimal polinomial of the inverse. Let the following be the original polynomial factored to complex terms of degree one: $$p(z)= \prod_{i=1}^4 (z -\lambda_i)$$ Then the minimal polinomial of $T^{-1}$ is $$q(z)= \prod_{i=1}^4 (z -\lambda_i^{-1})$$ This can be shown by observing that the inverse of a jordan block of the form $( \lambda, n) $ has jordan form $(\lambda^{-1},n)$ (1) and this extends to the fact that the jordan form of the inverse of a matrix is the same of the original matrix with all eigenvalues replaced by their inverse.