# GR: simple algebra

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:

$$g_{\bar{i}\bar{j}}=\Lambda_{\bar{i}}_{i}\Lambda_{\bar{j}}_{j} g_{ij}$$

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

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

thanks.

## Answers and Replies

CompuChip
Science Advisor
Homework Helper
If A and B are matrices, then the matrix product is given by
(AB)ik = Aij Bjk
by definition, and the transpose of A is given by
(AT)ij = Aji

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