Prove diagonalizabilty from a polynomial in $T$

Question

Suppose that $T^3 = 3T^2 - 2T$. Prove that T is diagonalizable.

I have proved that T is diagonalizable iff the minimal polynomial of T has no repeated roots.

$$T^3 - 3T^2 +2T = T(T^2-3T+2 ) = T( T-1)(T-2) $$ this polynomial has no repeated roots i would like to conclude that the minimal polynomial of $T$ cannot have any repeated roots? but im not sure theirs correlation between this polynomial and the minimal one? i think its just a notation thing cause my textbook and prof use diffrent notation for everything so im not really sure what $T^3 = 3T^2 - 2T$ is.

José Carlos Santos · Accepted Answer · 2018-03-05T17:54:02.433

3

The minimal polynomial of $T$ divided $x(x-1)(x-2)$ (since $T$ is a root of this polynomial) and all factors of this polynomial ($1$, $x$, $x-1$, $x-2$, $x(x-1)$, $x(x-2)$, $(x-1)(x-2)$, and $x(x-1)(x-2)$) have distinct roots.

edited Mar 05 '18 at 17:54

answered Mar 05 '18 at 17:45

José Carlos Santos

427,504

Thank you the since T is a root of this polynomial clarified my question – Faust Mar 05 '18 at 17:48
@Faust If my answer was useful, perhaps that you could mark it as the accepted one. – José Carlos Santos Mar 05 '18 at 18:00
it takes 10 minutes till you can accept the answer so i had to wait. – Faust Mar 05 '18 at 18:00

Prove diagonalizabilty from a polynomial in $T$

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