Inhomogeneous (poincare) lorentz transormation

[spacelike]

I'm reading a physics book and in the section on relativity they are using the Einstein summation convention, with 4vectors and matrices.

They say that the transformations take the form:
x^{\prime\mu}=x^{\nu}\Lambda^{\mu}_{\nu}+C^{\mu}
where it is required that \Lambda^{\mu}_{\nu} satisfy the following relation:
\eta_{\mu\nu}\Lambda^{\mu}_{\alpha}\Lambda^{\nu}_{\beta}=\eta_{\alpha\beta}
(note: I found the same thing on wikipedia, so you can see it in context if you like. http://en.wikipedia.org/wiki/Lorentz_transformation#Spacetime_interval it appears a tiny bit down from the section that the link takes you to.)

My problem is that this seems impossible to satisfy by my current understanding, but I know I must be wrong, I just cannot see how.

So we are summing over \mu and \nu in the above relation right? and we do this for all \alpha and \beta in order to satisfy all the components of the matrices.
My problem is what happens when we get to the following situation?:
\mu=0, \nu=1, \alpha=0, \beta=0
But, \eta_{01}=0, and \eta_{00}=-1. So there is no possible values of the \Lambda's that will satisfy this because we now have 0=-1, which is a contradiction.

Where did I go wrong with my thinking? Thanks.

 

[George Jones]

In an inertial coordinate system,
\eta_{\alpha\beta} = \eta_{\mu\nu}\Lambda^{\mu}_{\alpha}\Lambda^{\nu}_{ \beta} = -\Lambda^{0}_{\alpha}\Lambda^{0}_{ \beta} + \Lambda^{1}_{\alpha}\Lambda^{1}_{ \beta} + \Lambda^{2}_{\alpha}\Lambda^{2}_{ \beta} + \Lambda^{3}_{\alpha}\Lambda^{3}_{ \beta}

 

[Matterwave]

You're not summing, you've just assumed 4 values for the 4 variables. Remember you have to sum over mu and nu.

 

[spacelike]

Right! I knew it would have to have been something stupidly simple >.<

thanks guys.