Undergrad Matrix form of metric component transformation?

[farfromdaijoubu]

TL;DR

Question about the matrix form of a metric transformation from coordinate system to another.

The transformation of the metric from unprimed to primed coordinates is given in Moore's GR workbook as:
$g'_{uv} = \frac{\partial x^a}{\partial x^{'u}} \frac{\partial x^b}{\partial x^{'v}}g_{ab}$

In matrix form, by matching indices on both sides (so that u and v are the outer 'indices'), I got
$(g'_{uv}) = \left(\frac{\partial x^a}{\partial x^{'u}}\right)^T (g_{ab}) \left( \frac{\partial x^b}{\partial x^{'v}} \right)$


But on pg. 146 of Schutz's GR textbook, he writes the matrix form of the transformation as $$(g') = $(\Lambda) (g) \Lambda)^T$$, where $\Lambda = \left(\frac{\partial x^a}{\partial x^{u'}} \right)$.

How did he get this result? Did I misunderstand something? Because I don't think it's equivalent to what I got...

Thanks

 

[Orodruin]

Different books may have different definitions of which of the indices of $\partial x^a/\partial x^{u’}$ is to be interpreted as the row index.