Question

Define T:R^3 by
T(x,y,z)= (-x,x-y,3x+2y+z).
Check whether T satisfies the polynomial (x-1) (x+1)^2. Find the minimal polynomial of T.

Answer 1

Denoting by $I$ the identity operation, we have

$$(T + I) (x, y, z) = (0, x, 3x + 2y + 2z) \, , \\ (T + I)^2 (x, y, z) = (0, 0, 8x + 4y + 4z) \, , \\ (T - I) (x, y, z) = (- 2 x, x - 2 y, 3x + 2y) \, .$$

Combining these equations together, we obtain

$$(T - I) (T + I)^2 (x, y, z) \\ {} = (T - I) (0, 0, 8 x + 4 y + 4 z) = (0, 0, 0) \, .$$

Hence, $T$ satisfies the polynomial $p (x) = (x - 1) (x + 1)^2$. Considering $v = (1, 0, 0)$, we have

$$T (v) = (-1, 1, 3) \, , \quad T^2 (v) = (1, -2, 2) \, .$$

The vectors $v$, $T (v)$ and $T^2 (v)$ are linearly independent; hence, the minimal polynomial of $T$ has degree three. Since $p (x)$ has degree three and is a monic polynomial (its leading coefficient is equal to 1), and since $p (T) = 0$, it is the minimal polynomial.