Let \(T\) be a linear operator on the finite-dimensional space \(V\) with characteristic polynomial \[ f = (x - c_1)^{d_1} \cdots (x - c_k)^{d_k} \] and minimal polynomial \[ p = (x - c_1)^{r_1} \cdots (x - c_k)^{r_k}. \] Let \(W_i\) be the null space of \((T - c_iI)^{r_i}\).
Prove that \(W_i\) is the set of all vectors \(\alpha\) in \(V\) such that \((T - c_iI)^m\alpha = 0\) for some positive integer \(m\) (which may depend upon \(\alpha\)).
Prove that the dimension of \(W_i\) is \(d_i\). (Hint: If \(T_i\) is the operator induced on \(W_i\) by \(T\), then \(T_i - c_iI\) is nilpotent; thus the characteristic polynomial for \(T_i - c_iI\) must be \(x^{e_i}\) where \(e_i\) is the dimension of \(W_i\) (proof?); thus the characteristic polynomial of \(T_i\) is \((x - c_i)^{e_i}\); now use the fact that the characteristic polynomial for \(T\) is the product of the characteristic polynomials of the \(T_i\) to show that \(e_i = d_i\).)
Solution
(a).
1. Let \(S_i = \{\alpha \in V : (T - c_i I)^m \alpha = 0 \text{ for some } m \ge 1\}\) be the set of generalized eigenvectors for \(c_i\).
definition.
2. \(W_i = \ker(T - c_i I)^{r_i} \subseteq S_i\).
taking \(m = r_i\) shows every element of \(W_i\) is in \(S_i\).
3. Conversely, let \(\alpha \in S_i\), so \((T - c_i I)^m \alpha = 0\) for some \(m\).
take an arbitrary element of \(S_i\).
4. If \(m \le r_i\), then \((T - c_i I)^{r_i}\alpha = (T - c_i I)^{r_i - m}(T - c_i I)^m \alpha = 0\), so \(\alpha \in W_i\).
3.
5. If \(m > r_i\), then since the minimal polynomial is \(p = (x - c_1)^{r_1}\cdots(x - c_k)^{r_k}\), the polynomial \((x - c_i)^{r_i}\) divides \((x - c_i)^m\) only up to the factor \((x - c_i)^{r_i}\); more directly, \((T - c_i I)^m \alpha = 0\) implies \((T - c_i I)^{r_i}\alpha = 0\) because the minimal polynomial of \(T\) restricted to the generalized eigenspace for \(c_i\) is \((x - c_i)^{r_i}\).
the minimal polynomial of \(T\) on the generalized eigenspace for \(c_i\) is \((x - c_i)^{r_i}\), so \((T - c_i I)^{r_i}\) annihilates every generalized eigenvector for \(c_i\).
6. Hence \(S_i \subseteq W_i\), so \(W_i = S_i\).
2 and 5.
(b).
1. \(T_i = T|_{W_i}\) satisfies \((T_i - c_i I)^{r_i} = 0\), so \(T_i - c_i I\) is nilpotent.
1 (a) (every element of \(W_i\) is annihilated by \((T - c_i I)^{r_i}\)).
2. Hence the characteristic polynomial of \(T_i - c_i I\) is \(x^{e_i}\), where \(e_i = \dim W_i\).
a nilpotent operator on an \(e_i\)-dimensional space has characteristic polynomial \(x^{e_i}\) (its only eigenvalue is \(0\)).
3. Hence the characteristic polynomial of \(T_i\) is \((x - c_i)^{e_i}\).
2 (shifting by \(c_i\)).
4. \(V = W_1 \oplus \cdots \oplus W_k\) (primary decomposition), so the characteristic polynomial of \(T\) is the product of the characteristic polynomials of the \(T_i\).
primary decomposition theorem.
5. Hence \(f = (x - c_1)^{d_1}\cdots(x - c_k)^{d_k} = (x - c_1)^{e_1}\cdots(x - c_k)^{e_k}\).
3 and 4.
6. Therefore \(e_i = d_i\) for each \(i\), i.e. \(\dim W_i = d_i\).
5 (comparing exponents).
7. Q.E.D.
6 (a) and 6 (b).