## 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
$A= \left( \begin{array}{cc} c & 0 \\ 1 & c \end{array} \right)$
.