Possible Jordan canonical forms of identity matrix plus a nilpotent matrix I am working on the following Linear algebra problem: 
Suppose that $N$ is a nilpotent $5 \times 5$ real matrix (so $N^5$ is the zero matrix). List all possible Jordan canonical forms of $I + N$.
Here is where my thinking is at: I know how to list all possible Jordan canonical forms of a matrix, given its characteristic polynomial. I also know that, in this case, the characteristic polynomial of $N$ will be given by $p_N(x) = x^5$. However, I'm struggling with this problem because I don't know how to deduce from this what the characteristic polynomial of $I + N$ is. Is there a nice way to see what this is? 
I searched for if there is a nice relationship between a matrix's characteristic polynomial and the characteristic polynomial of the matrix plus the identity, but I couldn't find one. Does the fact that $N$ is nilpotent help at all to see what this is? 
Thanks!  

As you mentioned correctly- The characteristic polynomial of $N$ is $\chi_N(x)=x^5$, we know that not by computing, but by the nilpotent property and the dimension of $N$. To understand the characteristic polynomial of $N+I$ we need to go back to the definition: To any matrix $M$ it's characteristic polynomial is given by $\det(xI-M)$. Let's examine what happens when we add the identity matrix:
$$\det(xI-(N+I))=\det((x-1)I+N)$$
This is exactly the definition of the characteristic polynomial but instead of being a polynomial over of the variable $x$, it's shifted by $1$. It's a composition of the original characteristic polynomial with $x-1$, so if we had $\chi_N(x)=x^5$ then $\chi_{N+I}(x)=(x-1)^5$, and thus 1 is an eigenvalue of $N+I$.
Knowing all this we get these options of the jordan canonical form (up to rearranging etc.) (also, I'm using lower triangular matrices, I know some people use upper, these are equivalent):
$$
\left[\begin{array}{ccccc}{1} & {0} & {0} & {0} & {0} \\ {0} & {1} & {0} & {0} & {0} \\ {0} & {0} & {1} & {0} & {0} \\ {0} & {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {0} & {1}\end{array}\right],\left[\begin{array}{ccccc}{1} & {0} & {0} & {0} & {0} \\ {1} & {1} & {0} & {0} & {0} \\ {0} & {1} & {1} & {0} & {0} \\ {0} & {0} & {1} & {1} & {0} \\ {0} & {0} & {0} & {0} & {1}\end{array}\right],\left[\begin{array}{ccccc}{1} & {0} & {0} & {0} & {0} \\ {1} & {1} & {0} & {0} & {0} \\ {0} & {1} & {1} & {0} & {0} \\ {0} & {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {0} & {1}\end{array}\right],\left[\begin{array}{ccccc}{1} & {0} & {0} & {0} & {0} \\ {1} & {1} & {0} & {0} & {0} \\ {0} & {1} & {1} & {0} & {0} \\ {0} & {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {1} & {1}\end{array}\right]
$$
$$
\left[\begin{array}{ccccc}{1} & {0} & {0} & {0} & {0} \\ {1} & {1} & {0} & {0} & {0} \\ {0} & {0} & {1} & {0} & {0} \\ {0} & {0} & {1} & {1} & {0} \\ {0} & {0} & {0} & {0} & {1}\end{array}\right],\left[\begin{array}{ccccc}{1} & {0} & {0} & {0} & {0} \\ {1} & {1} & {0} & {0} & {0} \\ {0} & {0} & {1} & {0} & {0} \\ {0} & {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {0} & {1}\end{array}\right],\left[\begin{array}{ccccc}{1} & {0} & {0} & {0} & {0} \\ {1} & {1} & {0} & {0} & {0} \\ {0} & {1} & {1} & {0} & {0} \\ {0} & {0} & {1} & {1} & {0} \\ {0} & {0} & {0} & {1} & {1}\end{array}\right]
$$
