# Real Canonical Form of Arbitrary Size Matrices

I have been searching the site and the Mathematica documentation, but have not found anything regarding this.

If we find the Jordan Form of the following matrix, we get complex values, but I would like to find the real canonical form (all values in the Jordan matrix are real).

 JordanDecomposition[{{3, -2}, {1, 1}}]


Using this decomposition (or MatrixExp), we are able to find the exponential, $e^{t A}$, of this matrix.

For this particular problem, we find the real canonical form as:

$$P = \left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \\ \end{array} \right)$$

$$J_R = P^{-1} A P = \left( \begin{array}{cc} 2 & 1 \\ -1 & 2 \\ \end{array} \right)$$

We can then use this real variant to find the matrix exponential as:

$$e^{t A} = P e^{t J_R} P^{-1}$$

You can see another $2\times 2$ and a $3\times3$ example here and here is a larger example.

Is Mathematica able to find the Real Canonical Form for an arbitrarily sized matrix (hopefully I am just missing something basic)?

Update

These notes provide an algorithmic type approach.