Simplifying GR: Algebraic Solutions Explained

[proguinn]

This isn't homework but since it's a simple algebra thought I suppose it's the most appropriate forum- I'm working through some basic general relativity and just came across a line i think I'm being stupid about:

<br /> g_{\bar{i}\bar{j}}=\Lambda_{\bar{i}}_{i}\Lambda_{\bar{j}}_{j} g_{ij}<br />

then, it's said that it's obvious that this means:
<br /> g_{\bar{i}\bar{j}}= (\Lambda^T g \Lambda)_{\bar{i}\bar{j}}<br />

Is there an easy way to show this, other than by inspection?

thanks.

 

[CompuChip]

If A and B are matrices, then the matrix product is given by
(AB)_ik = A_ij B_jk
by definition, and the transpose of A is given by
(A^T)_ij = A_ji

From that it easily follows (just write them in the correct order, then swap the indices on one of them creating a transpose).