## The Jordan Form

Jordan block matrices are square matrices with entries from a field\[F\]

along the leading diagonal all equal to each other. Above this diagonal is a diagonal of 1s.The entries along the leading diagonal may be real or complex. The following matrices are Jordan Block matrices.

\[ \left( \begin{array}{cc} i & 1 \\ 0 & i \\ \end{array} \right), \: \left( \begin{array}{ccc} 2+i & 1 & 0 \\ 0 & 2-i & 1 \\ 0 & 0 & 2-i \end{array} \right) \]

The following matrices are not Jordan Block matrices.

\[ \left( \begin{array}{cc} i & 1 \\ 0 & 2i \\ \end{array} \right), \: \left( \begin{array}{ccc} 2+i & 1 & 1 \\ 0 & 2-i & 1 \\ 0 & 0 & 2-i \end{array} \right) \]

The Jordan form of a matrix consists of a matrix having Jordan Block submatrices along the leading diagonal, and all other entries zero.

The matrix

\[ \left( \begin{array}{cccccc} 4 & 0 & 0 & 0 & 0 & 0 \\ 0 & 3 & 1 & 0 & 0 & 0 \\ 0 & 0 & 3 |& 1 & 0 & 0 \\ 0 & 0 & 0 & 3 & 1 & 0 \\ 0 & 0 & 0 & 0 & 2-i & 1 \\ 0 & 0 & 0 & 0 & 0 & 2-i \end{array} \right) \]

is an example.