How to prove the changing of basis matrices is unitary matrices

Question

Suppose A is matrix defined on C
I need to prove that A is written as A=OTO*
T is triangular matrix
O* is Conjugate transpose

Is it enough (and how) if I proved that O is unitary that mean O* = (O inverse) and that just mean I proved A is traingularizable because its distinct polynomial is divided

Answer 1

This is called the Schur decomposition. See Schur decomposition. For constructive proofs by induction, see the lecture notes here or here.