# Find the invariant factors and elementary divisors from the relations matrix.

Let $V$ be a finite dimensional vector space over $\Bbb C$, and $T$ be a linear operator on $V$. Consider $V$ as an $\Bbb C[x]$-module by defining $xv = T(v)$ for each $v \in V$.

Let $$A = \begin{bmatrix} x^2(x - 1)^2 & 0 & 0\\ 0 & x(x - 1)(x - 2) & 0 \\ 0 & 0 & x(x - 2)^2 \end{bmatrix}$$

be the relation matrix for $V$ with respect to {$v_1, v_2, v_3$}.

1) List the invariant factors and elementary divisors of $V$.

Attempt: In general, I would reduce the relation matrix to smith normal form to find the invariant factors, and from there I could find the elementary divisors. However, this matrix is hard to reduce to SNF, so I think they want me to find another approach.

From the relation matrix I believe $$V = \frac{F[x]}{x^2(x - 1)^2} \oplus \frac{F[x]}{x(x - 1)(x-2)} \oplus \frac{F[x]}{x(x - 1)^2}$$ and the elementary divisors are $x^2, (x-2)^2, x, x, (x-1)^2, (x - 1) (x - 2)$, which would mean the cyclic decomposition is

$$V=\frac{F[x]}{x^2(x - 1)^2(x-2)^2} \oplus \frac{F[x]}{x(x - 1)(x-2)} \oplus \frac{F[x]}{x}$$ with invariant factors $x^2(x - 1)^2(x-2)^2, x(x - 1)(x-2), x$.

2) Describe the decomposition of $V$ by the primary decomposition theorem, and find the Jordan form of the operator $T$.

Attempt: From the invariant factors, I believe the minimal polynomial $p = x^2(x - 1)^2(x-2)^2$, and the characteristic polynomial is $f = x^2(x - 1)^2(x-2)^2x(x - 1)(x-2)x$. Hence if $N_1 = \mathrm{null}(T^2), N_2 = \mathrm{null}(T - I)^2, N_3 = \mathrm{null}(T- 2I)^2$, then $V=N_1 \oplus N_2 \oplus N_3$ is the primary decomposition. I'm not sure, but I think the Jordan form will be $$A= \begin{bmatrix} 1 & 0 & 0 & 0 &0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & B \\ \end{bmatrix}$$ for some block matrix $B$.

Can't we just determine the Jordan form from the elementary divisors? $$A = \begin{bmatrix} 0&&&&&&&&&\\ &0&&&&&&&&\\ &&0&&&&&&&\\ &&1&0&&&&&&\\ &&&&1&&&&&\\ &&&&&1&&&&\\ &&&&&1&1&&&\\ &&&&&&&2&&\\ &&&&&&&&2&\\ &&&&&&&&1&2 \end{bmatrix}$$