It is true that if an upper triangular matrix $A$ with complex entries has distinct elements on the diagonal, then $A$ is diagonalizable. However, I don't think the converse is true. Is there a complete characterization of all diagonalizable upper triangular matrices?
[Math] Diagonalizable upper triangular matrices
linear algebra
Best Answer
Every upper triangular matrix with distinct elements on the diagonal is diagonalizable, because $$\det(A-\lambda I)=\prod_{i=1}^n (a_{ii}-\lambda)$$ with $a_{ii}\neq a_{jj}$ for $i\neq j$, so every eigenvalue has multiplicity $1$.
The converse is not true. Take $A=I$. Then $A$ is diagonalized, but not with distinct values on the diagonal.