Show that a matrix is a Lorentz transformation

[fineTuner]

Homework Statement

Given the matrix
$$ \Omega = \begin{pmatrix}
0 & -\psi & 0 & 0 \\
-\psi & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix}$$
show that ## e^{\Omega}## is a Lorentz transformation along the x-axis with ## \beta = tanh(\psi)##

Homework Equations

During the lesson we derived from the standard Lorentz transformation matrix the following matrix, where ##\psi## is the rapidity:

$$ \Lambda = \begin{pmatrix}
cosh(\psi) & -sinh(\psi) & 0 & 0 \\
-sinh(\psi) & cosh(\psi) & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix} $$

Other equations:
##cosh(\psi)=\gamma##
##sinh(\psi)=\gamma \beta##

The Attempt at a Solution

[/B]
From ## \beta = tanh(\psi)##:
## \psi=arctg(\beta) = \ln\sqrt{\frac{1+\beta}{1-\beta}} ##
## e^{-\psi} = \sqrt{\frac{1-\beta}{1+\beta}}##
I think i have to show that the two matrices (##\Lambda## and ##e^{-\Omega}##) are the same, but i can't understand why there are zeros on the diagonal. For the two first zeros on the diagonal ##cosh(\psi)=0##, so ##\psi = \frac \pi 2##.
I think there's an error somewhere, because with the previous formulas it turns out that ##\beta = 1## and ##v=c##.

To be honest, i can't find the right way to solve the problem, maybe it's just algebra? Can you please give me a hint? Thank you!

 

[PeroK]

What is the definition of ##e^{\Omega}##?

Can you calculate ##\Omega^n##?

 

[fineTuner]

##\Omega^n## can be obtained multiplying the matrix n times. I underestimated the definition of ##e^{\Omega}##, i guess i can't obtain it taking the exponential of each element... now I'm reading the definition on wikipedia.

 

[nrqed]

##\Omega^n## can be obtained multiplying the matrix n times. I underestimated the definition of ##e^{\Omega}##, i guess i can't obtain it taking the exponential of each element... now I'm reading the definition on wikipedia.

The exponential of a matrix is defined as in quantum mechanics, through its Taylor expansion (here around ##\psi=0##).

 

[robphy]

Here's a variation of this problem:
https://www.wolframalpha.com/input/...)+{{0,-q},{q,0}}^2/(2!)+{{0,-q},{q,0}}^3/(3!)
Note the pattern of powers of the given matrix, before combining terms.