Jordan Normal Form physical applications

[Maybe_Memorie]

Are there any applications of the Jordan Normal Form of a matrix in physics?
If so, please explain?

 

[micromass]

The Jordan Normal Form is very often used to solve linear differential equations. Such an equations can often be represented in the form

$$y^\prime=Ay,~y(0)=y_0$$

with A a matrix. The solution to such an equation is $$y(t)=e^{At}y_0$$. The problem is to calculate the matrix exponential $$e^{At}$$, this can be very difficult. To ease the problem, we calculate the Jordan Normal Form of A. In this from, calculating the matrix exponential becomes very easy!
The fun part is of course to calculate the Jordan Normal Form, as this can be quite difficult...

 

[HallsofIvy]

A slight variation on what micromass said:
If you have the d.e. y'= Ay, and B is such that $B^{-1}AB= J$, the "Jordan form" for A, then we can multiply both sides of the equation by $B^{-1}$:
$$(B^{-1}y)'= B^{-1}Ay= B^{-1}A(BB^{-1})y= (B^{-1}AB)(B^{-1}y)$$
and, letting $x= B^{-1}y$ write the equation as
[tex]x'= Jx[/itex].

Now, since J is a Jordan form matrix, that gives a system of "almost uncoupled" equations. If we write x as a column matrix
$$\begin{bmatrix}x_1 \\ x_2 \\ ---\\ x_n\end{bmatrix}$$

Then every equation is of the form $x_k'= \lambda_kx_k$ or $x_k'= \lambda_nx_k+ x_{k+1}$. The very last equation is, of course, [itex]x_n'= \lambda_n x_n[itex], because there is no "$x_{n+1}$" and is easy to solve. Then, if $x_{n-1}'= \lambda_{n-1}x_{n-1}+ x_n$, because we already have $x_n$, that is easy to solve, etc.

Finally, since $x= B^{-1}y$, $y= Bx$.

If a matrix happens to be "diagonalizable" (all those "1" above the diagonal are unecessary), we can completely "uncouple" all those equations, whether differential equations or not, and have, basically, n separate, simpler, problems. If the matrix is NOT diagonalizable, we can still simplify as much as possible using the "Jordan Normal Form".