If we know the size of Jordan blocks of a matrix, what else can we derive?

Question

If we have a 4x4 matrix with two distinct eigenvalues and if all Jordan blocks are of size 1x1, can we say that the matrix must be diagonal?

I am struggling to correlate the number non-diagonal elements with Jordan block sizes. Diagonal matrix is an option; but, are there any other possibility? What can we say about the matrix?

Answer 1

You cannot (ever, unless the Jordan form is a scalar multiple of the identity, in which case it is equal to the matrix) conclude from the Jordan form whether a square matrix is diagonal. The best thing you might deduce from the Jordan is that the matrix is diagonalisable, which means precisely that all Jordan blocks are $1\times1$.