-1

I encountered this question and I need some assistance to solve it.

$T$ is a linear transformation from $V \to V$

We are given that $T^2 = T$

Show that $T$ can be diagonalized.

Daniel Robert-Nicoud
  • 30,663
Oria Gruber
  • 13,035
  • I know that, but who said that T is 2x2? Just because there are 2 distinct roots does not mean it can be diagonalized. what if T is 3x3? in that case you need 3 roots... – Oria Gruber Oct 27 '13 at 21:15
  • 1
    similar to: http://math.stackexchange.com/questions/73862/diagonalization-of-a-projection – Katsu Oct 27 '13 at 21:21
  • @Katsu It's not only similar, it's the same – egreg Oct 27 '13 at 21:24
  • @egreg: Sorry it's not my native language. It's what I meant. – Katsu Oct 27 '13 at 21:25
  • Regarding your deleted question: $x\equiv a!\pmod {!n}!\iff! n=n_1n_2\cdots n_k\mid x-a$

    $!\iff!n_1,n_2,\ldots,n_k\mid x-a!\iff!\forall n_i\ (x\equiv a!\pmod{!n_i})$

    Remember $\ a,b\mid n!\iff! ab\mid n\ $ given $(a,b)=1$,

    which is true because $\ a,b\mid c!\iff! \text{lcm}(a,b)\mid c\ $ is the definition of $\text{lcm}$

    and $\text{lcm}(a,b)=\frac{ab}{(a,b)}$.

     – user26486 May 05 '15 at 23:17

1 Answers1

0

Hint: Assume it can't, then you can still take it in Jordan's normal form. Derive a contradiction using the condition $T^2=T$.

Daniel Robert-Nicoud
  • 30,663