Why a triangular matrix can be non-diagonalizable

Question

We can turn every triangular matrix into a diagonal matrix with elementary Matrices.

So, How can it be possible to have a matrix which is similar to a triangular matrix but not to a diagonal matrix?

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In addition, what is the best trick to know if a matrix is diagonalizable or not?

Answer 1

you have only one condition that is necessary and sufficient condition to know whether your matrix is diagonalizable or not.and that is iff your minimal polynomial is product of non repeated factors in the field concerned.i.e all the roots must lie in the field itself and their multiplicities should be 1 in the minimal polynomial. for the first part take any matrix upper or lower triangular matrix with 0 on the main diagonal.then you see that the matrix is not diagonalizable but triangulizable