Explicit and Implicit Matrix Notation Question

[Alex86]

Ok so this is a fairly stupid question I'm sure, but I'm not quite clear about the following:

Given a Lorentz transformation we require the following to hold:

$$g_{\sigma\rho}\Lambda^{\sigma}_{\mu}\Lambda^{\rho}_{\nu} = g_{\mu\nu}$$

In other notation this is written:

$$\Lambda^{T}g\Lambda = g$$

where $$\Lambda \in O(1,3)$$ and g is the Minkowski metric.

The first use of tensor notation is fine, however I am unsure why in the second expression the first $$\Lambda$$ is transposed?

Any help greatly appreciated,
Alex

 

[slider142]

In the first sum over sigma, the "rows" of g are summed with the "rows" of lambda. In matrix multiplication, rows are summed with columns. If we write lambda and g as matrices and want to express the same product as a matrix multiplication, then in order for the correct components to be multiplied, we must transpose lambda.