Let A be a n times n matrix that is lower triangular Prove t

Let A be a n times n matrix that is lower triangular. Prove that if the diagonal entries distinct then A is diagonalizable.

Solution

Since A is lower triangular matrix with diagonal entries all distinct, then it has n distinct eigenvalues,

that is, each eigenvalue has multiplicity 1. Then we have n LI eigenvectors, therefore A is diagonalizable.