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

TheoremEvery 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 \[A= \left( \begin{array}{cc} c & 0 \\ 1 & c \end{array} \right) \]

.