Lagrangian: kinetic matrix Z_ij and mass matrix k_ij

[GGGGc]

Can somebody explain why the kinetic term for the fluctuations was already diagonal and why to normalize it, the sqrt(m) is added? Any why here Z_ij = delta_ij?
Quite confused about understanding this paragraph, can anybody explain it more easily?

 

[vanhees71]

How can we answer this question, without knowing, what the example actually is? It's also mandatory to quote the textbook/paper you picture is taken from.