Necessary and sufficient condition for diagonalizabilty of rank $1$ $n\times n$ square matrix. Let $A$ be a square matrix $n\times n$ of rank $1$. I want to find a necessary and sufficient condition for the diagonalizable of this particular form of $A$.

I know equivalent conditions for diagonalizability of $n\times n$ matrix $A$.

*

*Its minimal polynomial has no repeated roots.

*It has $n$ linearly independent eigenvectors.

I found interesting posts about the full rank of $A$ in Diagonalizable vs full rank vs nonsingular (square matrix) ....,

I think it is possible to formulate such a criterion.

A nonzero square matrix of rank $1$ is diagonalizable if and only if
its trace is not equal to $0$.

The trace of a square matrix is the sum of its diagonal elements.
