What are some properties or characteristics of singular eigenvector matrices?

Question

I'm trying to predict when a matrix $A$ will have an eigenvector matrix $X$ that is singular. What properties or characteristics would $A$ or $X$ have to make $X$ singular??

Answer 1

A square matrix $A$ can only have a non-singular eigenvector matrix if $A$ is diagonalizable.

Strictly speaking, every matrix size $2$ and bigger has a singular "eigenvector matrix".

EDIT: to your comment: if we have an $n \times n$ matrix $A$ whose eigenvalues are $\{\lambda_1,\dots,\lambda_k\}$, where this list repeats no eigenvalues, then $A$ is diagonalizable if and only if $$ (A - \lambda_1 I)(A - \lambda_2 I)\cdots(A - \lambda_k I) = 0 $$ where $I$ is the identity matrix. This uses the idea of a matrix's minimal polynomial.