Let \(V\) be a finite-dimensional vector space. What is the minimal polynomial for the identity operator on \(V\)? What is the minimal polynomial for the zero operator?
Solution
1. The identity operator \(I\) satisfies \(I - I = 0\), i.e. the polynomial \(x - 1\) annihilates \(I\).
\((x-1)(I) = I - I = 0\).
2. No nonzero polynomial of degree \(0\) annihilates \(I\).
a constant \(c\) annihilates \(I\) only if \(cI = 0\), forcing \(c = 0\).
3. Hence the minimal polynomial of \(I\) is \(x - 1\).
1 and 2 (the minimal polynomial is the monic annihilating polynomial of least degree).
4. The zero operator \(0\) satisfies \(x(0) = 0\), i.e. the polynomial \(x\) annihilates it.
\(x(0) = 0\).
5. No nonzero constant annihilates the zero operator.
\(c \cdot 0 = 0\) for all \(c\), but the minimal polynomial must be monic and nonzero; the least-degree monic annihilating polynomial is \(x\).
6. Hence the minimal polynomial of the zero operator is \(x\).
4 and 5.
7. Q.E.D.
3 and 6.