Smith normal form of a polynomial matrix

Question

I have the following matrix

$$P(s) := \begin{bmatrix} s^2 & s-1 \\ s & s^2 \end{bmatrix}$$

How does one compute the Smith normal form of this matrix? I can't quite grasp the algorithm.

Answer 1

One easy way is by looking at fitting ideal.

If \begin{bmatrix} d_1 & 0\\ 0 & d_2 \end{bmatrix} is the required smith normal form then,

check that , $Fitt_1(P) = (d_1)=(1)$ and $Fitt_2(P) = (d_1d_2)=(x^4-x^2+x)$.