I want to solve this question:
Let $A$ be an $n \times n$ matrix over a field $k,$ all of whose columns are the same. Describe the conditions under which $A$ is diagonalizable. Justify your answer.
I saw this question Diagonalizability and linear independence of columns here, but I do not know what is the relation between minimal polynomials and diagonalizability and columns, could someone explain this to me please?
Does linear independence of columns imply diagonal matrix but the reverse is not true? how this help me in answering my question above?
Hint: Note that the rank of such a matrix is either $0$ or $1$. What does that tell you about the eigenvectors of this matrix? From there, consider the trace of the matrix and what this trace tells you about the matrix's eigenvalues.