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Homework Help: Linear Operators, characteristic polyn.

  1. Mar 8, 2006 #1

    JFo

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    Hi, I'm a little stuck on this problem. The question is:

    Let [itex] T [/itex] be a linear operator on a two dimensional vector space [itex] V [/itex], and suppose that [itex] T \neq cI [/itex] for any scalar c. (here [itex] I [/itex] denotes the identity transformation). Show that if [itex] U [/itex] is any linear operator on [itex] V [/itex] such that [itex] UT = TU [/itex], then [itex] U = g(T) [/itex] for some polynomial [itex] g(t) [/itex]

    This is what I have so far:

    Let f(t), g(t), h(t) denote the characteristic polynomial of T, U, and UT respectively. Since V is two dimensional, we know that f, g, and h are of degree 2 with leading coefficient (-1)^2 = 1. Therefore there exists scalars [itex] a_0, a_1, b_0, b_1, c_0, c_1 [/itex] such that

    [tex] f(t) = t^2 + a_1 t + a_0 [/tex]
    [tex] g(t) = t^2 + b_1 t + b_0 [/tex]
    [tex] h(t) = t^2 + c_1 t + c_0 [/tex]

    from the Cayley-Hamilton Theorem, we know that

    [tex] f(T) = T^2 + a_1 T + a_0 I = T_0 \ \ \ \ (1)[/tex]
    [tex] g(U) = U^2 + b_1 U + b_0 I = T_0 \ \ \ \ (2)[/tex]
    [tex] h(UT) = (UT)^2 + c_1 UT + c_0 I = T_0 \ \ \ \ (3)[/tex]

    where [itex] T_0 [/itex] denotes the zero transformation

    [tex] (1) \Rightarrow T^2 = -a_0 I - a_1 T \ \ \ \ (4) [/tex]
    [tex] (2) \Rightarrow U^2 = -b_0 I - b_1 U \ \ \ \ (5) [/tex]
    [tex] (3) \Rightarrow (UT)^2 = -c_0 I - c_1 UT \ \ \ \ (6) [/tex]

    composing (5) with (4) we get

    [tex] U^2 T^2 = (UT)^2 = (-b_0 I - b_1 U)(-a_0 I - a_1 T) [/tex]
    [tex] \Rightarrow (UT)^2 = a_0b_0 I + a_0b_1 U + a_1b_0 T + a_1b_1 UT \ \ \ \ (7)[/tex]

    comparing (6) and (7) we have

    [tex] -c_0 I - c_1 UT = a_0b_0 I + a_0b_1 U + a_1b_0 T + a_1b_1 UT \ \ \ \ (8)[/tex]

    This is where I'm stuck,,, I would like to be able to compare the c coefficients witht the a,b coefficients to come up with an equation for U in terms of I and T, but I'm not sure how I'm supposed to use the fact that [itex] T \neq cI [/itex] for any scalar c.

    I realize that reading through this is a major pain in the butt, but thanks for your help (really!).
     
    Last edited: Mar 8, 2006
  2. jcsd
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