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Proof That All 2 x2 Complex Matrices are Similar to a Diagonal or Lower Triangular Matrix

Theorem
Every 2 x 2 matrix over the field of complex numbers is similar to a diagonal or lower triangular 2 x 2 matrix.
Proof
If  
\[T\]
  is a linear operator for which tho characteristic Polynomial factors completely over the scalar field, then there is an Ordered basis for  
\[V\]
  in which  
\[T\]
  is represented by a matrix which is in Jordan form. The characteristic polynomial for the given operator is  
\[(x-c_1)(x-c_2)\]
  where the eigenvalues  
\[c_1, \: c_2\]
  are complex numbers. If the eigenvalues are distinct, then the eigenvectors associated with these eigenvalues form a basis for  
\[\mathbb{C}^2\]
. With respect to this basis,  
\[T\]
  can be represented by a diagonal matrix  
\[A= \left( \begin{array}{cc} c_1 & 0 \\ 0 & c_2 \end{array} \right) \]
.
If  
\[c_1=c_2=c\]
  then the minimum polynomial is either  
\[(x-c)\]
  or  
\[(x-c)^2\]
.
If  
\[(x-c)\]
  then the Jordan canonical form of the matrix representing  
\[T\]
  is  
\[A= \left( \begin{array}{cc} c & 0 \\ 0 & c \end{array} \right) \]
.
If  
\[(x-c)^2\]
  then the Jordan canonical form of the matrix representing  
\[T\]
  is  
\[A= \left( \begin{array}{cc} c & 0 \\ 1 & c \end{array} \right) \]
.

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