The grade of a start vector and its relative minimal polynomial

Let $A$ be a square matrix over $\mathbb{R}$ or $\mathbb{C}$, and let $b$ be a vector of matching size. The relative minimal polynomial of $b$ with respect to $A$ is the unique monic polynomial over the same field $q_{A,b}$ of least degree such that

$q_{A,b}(A)b=0.$

Its degree

$\nu (A,b):=\mathrm{deg}{q}_{A,b}$

is the grade of $b$ relative to $A$.

If $b=0$, then $q_{A,b}=1$ and $\nu (A,b)=0$.

This polynomial is well defined. If $A$ is $n\times n$, then the $n+1$ vectors $b,Ab,\dots ,A^{n}b$ are linearly dependent, so some nonzero polynomial annihilates $b$; dividing a least-degree annihilator by its leading coefficient makes it monic. If two monic least-degree annihilators existed, their difference would be an annihilator of strictly smaller degree, so they would be equal.