Elementary divisors of $p(A)$. 
Let $\mathbb K$ be a field and $A\in M_n(\mathbb K)$. Consider a polynomial $p(x)\in\mathbb K[x]$. How are the elementary divisors of $A$ and the elementary divisors of $p(A)$ related?

 A: Notice that for any finite dimensional $K$-vector space $V$, and any linear map $A:V\rightarrow V$, and $p(x)\in K[x]$, the characteristic polynomial of $p(A)$ is 
$$\prod_{\lambda} (x-p(\lambda)),$$
where the $\lambda$ runs over the eigenvalues of $A$ counted with multiplicities. 
Similarly, assume that for a set of $\lambda$'s(not necessarily eigenvalues), a polynomial $\prod_{\lambda} (x-\lambda)\in K[x]$, then $\prod_{\lambda} (x-p(\lambda))\in K[x]$. (This can be done either by considering Galois group of splitting field of the former polynomial, or by considering symmetric functions) 
Now, regard the vector space $K^n$ as $K[x]$-module with $x$-action as $A$-multiplication. Then decompose this using the structure theorem over PID: 
$$K^n \simeq \bigoplus_{i\leq r} K[x]/(p_i),$$
where each $p_i$ is a power of irreducible polynomial over $K$. Then we have $\{p_i\}_{i\leq r}$ as elementary divisors of $A$. 
This allows an invariant space decomposition of $K^n$ as 
$K^n\simeq \bigoplus _{i\leq r} V_i$, with each $V_i\simeq  K[x]/(p_i)$ is $A$-invariant, so it is also $p(A)$-invariant. Thus, we can consider $p(A)$ as linear map on $V_i$ for each $i$. 
On each $V_i$, the minimal polynomial of $A$ coincides with characteristic polynomial, which is $p_i$. Let $p_i(x)=\prod_{\lambda\in E_i}(x-\lambda)$ (where $E_i$ denotes the eigenvalues of $A$ restricted to $V_i$ counted with multiplicity). Then the characteristic polynomial of $p(A)$ restricted to $V_i$ will be 
$$q_i(x)=\prod_{\lambda\in E_i} (x-p(\lambda)),$$
and this will also be in $K[x]$. 
Regard $V_i$ as $K[y]$-module where $y$-action is $p(A)$-multiplication. Then since $p(A)$ has characteristic polynomial $q_i(x)$, the structure of $V_i$ as $K[y]$-module will be 
$$\bigoplus_{j\leq t_i} K[y]/(q_{i,j}),$$
where each $q_{i,j}$ is a power of irreducible polynomial which divides $q_i$. 
We combine these together, then $K^n$ as $K[y]$-module has a structure:
$$\bigoplus_{i\leq r} \bigoplus_{j\leq t_i} K[y]/(q_{i,j}).$$
Then we have $\{q_{i,j}|i\leq r, j\leq t_i\}$ as elementary divisors of $p(A)$. 
Hence, what we can say in brief is that: 
The elementary divisors of $p(A)$ consists of some powers of irreducible polynomials which divide $\prod_{\lambda} (x-p(\lambda))$, where $\lambda$ runs over eigenvalues of $A$ counted with multiplicity. 
