I am currently reading Jim Hefferon's Linear Algebra.
In chapter 5, nilpotence, strings, he goes through the process of finding a string basis of a map, and proves that there exists a string basis for any nilpotent transformation. He then suddenly states that from this string basis, you can create a matrix consisting of only 0's, except subdiagonal 1's in blocks. Also this block-diagonal form, is the canonical form of similar nilpotent matrices. He doesn't go into many details, and latter on the exercises, he says that the canonical form can be "immediately" found just by knowing the amount of strings and their length (not even knowing the actual basis vectors).
So my two questions: 1) Is it obvious that the canonical form is a block-diagonal? Or does it need a rigorous proof that the text simply ignores? 2) How do we find the canonical form, if we know the strings and their lengths?