How exactly does one go about finding the minimal polynomial of a matrix?
So here I have the matrix whose first row is $(0,0,1)$ and second and third rows are $(0,0,0)$. I have worked out the characteristic equation to be $x^3$ but how would I find the minimal polynomial from here?
Any help would be much appreciated.
Best Answer
The minimal polynomial of a square matrix $A$ is the monic polynomial $p$ of lowest degree such that $p(A) = 0$. If $q$ is another polynomial such that $q(A) = 0$, then $p$ is a factor of $q$. In particular, by the Cayley-Hamilton Theorem, the characteristic polynomial of $A$ is such a $q$, so the minimal polynomial divides the characteristic polynomial.
In this case, the characteristic polynomial is $q(x) = x^3$. The only monic polynomials which divide $q$ are $p_0(x) = 1$, $p_1(x) = x$, $p_2(x) = x^2$, and $p_3(x) = x^3$. Clearly, $p_0(A) \neq 0$, so the minimal polynomial is one of $p_1$, $p_2$, $p_3$. You can check which of these satisfy $p_i(A) = 0$. Choosing the polynomial of lowest degree is the minimal polynomial for $A$.