16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 4

Elementary Canonical Forms

Mathematics — Linear Algebra · primary-decomposition-characteristic-polynomial

Problem

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_i I)^{r_i}

. (a) Prove that

W_i

is the set of all vectors

\alpha

V

such that

(T - c_i I)^m \alpha = 0

for some positive integer

m

(which may depend upon

\alpha

). (b) Prove that the dimension of

W_i

d_i

. (Hint: If

T_i

is the operator induced on

W_i

T

, then

T_i - c_i I

is nilpotent; thus the characteristic polynomial for

T_i - c_i I

must be

x^{e_i}

where

e_i

is the dimension of

W_i

(proof?); thus the characteristic polynomial of

T_i

(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