This is an elementary question which is do-able by hand but I am actually looking for suggestions or book references since I am sure that someone did this somewhere:
suppose $ A = \left( \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{array} \right) \in M_2(\mathbb{C}). $
Find $g=(g_{ij})\in GL(2,\mathbb{C})$ so that $gAg^{-1}$ is upper triangular.
One method is to explicitly write down $gAg^{-1}$ and set the function in 2nd row, 1st column equal to zero (which is $-a_{12} g_{21}^2 + g_{22} (a_{11} g_{21} - a_{22} g_{21} + a_{21} g_{22}) = 0 $) and attempt to find $g$ this way, while a second method is to find the eigenvalues of $A$ (the two eigenvalues may or may not be distinct) and find their eigenvectors.
Wiki recommends Linear Algebra Done Right by Sheldon Axler and I think Sheldon proves that any $A\in M_n(\mathbb{C})$ can be put into an upper triangular form using induction.
Either of the methods that I mentioned above seems to be quite messy if I want to explicitly write down such $g$ for any $A\in M_2(\mathbb{C})$, or even for any $A\in M_n(\mathbb{C})$.
Do you have any recommended approach or references because I would like to explicitly write down $g\in GL(n,\mathbb{C})$ so that $gAg^{-1}$ is upper triangular, where $A\in M_n(\mathbb{C})$.