Why In Matrix Triangulation The Eigenvalues Are On The Diagonal

Question

I understand that we take the eigenvalue and corsponding eigenvector, completing to a basis and when the matrix is multiply by the eigenvector we will get the eigenvalue, but if we have a multiplicity of $2$ and one eigenvector, why we will get the eigenvalues twice on the diagonal?

Answer 1

A hint:

Assume you are given a triangular matrix and want to compute its eigenvalues via the characteristic polynomial. What would you do?