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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 $A\in M_n(\mathbb{C})$.

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    (A little reflection shows that if you could find a $g$ whose entries are, say, rational functions on the entries of a generic $n$-by-$n$ matrix $A$, then you could then find formulas for the roots of polynomials of degree $n$ which are rational functions on the coefficients of the polynomial, and so on. We know such a feat is simply not possible)2012-06-30

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An algorithm for finding $g$, with the added condition of taking $g$ to be unitary, is given in Hogben's Handbook of linear algebra.

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    Thanks Jonas! I'll take a look at that. =)2012-06-30