# How to find the 'real' jordan canonical form of a matrix

Given that the the Jordan normal form of a matrix is,

$J=\begin{bmatrix}2&1&0&0\\0&2&0&0\\0&0&1-i&0\\0&0&0&1+i\end{bmatrix}$

How do you find the 'real' canonical form of the matrix?

• You could get a real $2 \times 2$ block insted of the complex diagonal block. This would also give you real basis-vectors – Laray Mar 29 '17 at 8:09

Wubbish. There clearly is such a thing. Just replace the complex pair by the $2\times 2$ block below.
$$J=\left[\begin{array}{rrrr}2&1&0&0\\0&2&0&0\\0&0&1&-1\\0&0&1&1\end{array}\right]$$
$$\left[\begin{array}{rr}a&-b\\b&a\\\end{array}\right]$$ which can be used to represent arbitrary complex numbers $a+bi$ or $a-bi$ as long as you are consistent on which you are using for each new number / block.