I am reading a paper and I am not so sure about some things. I will explain the setup-
We suppose $(Y,\mu)$ is a probability space with a metric $d_Y$ and $G$ is a compact connected Lie group. Suppose further that $\mathbb{R}^d$ is a representation of $G$, and the image of the representation $G\to GL_d$ lies in the orthogonal group $O_d$.
The paper says that the Euclidean metric on $\mathbb{R}^{d\times d}$ restricts to a pseudometric on $G$.
First of all, what is the Euclidean metric on $\mathbb{R}^{d\times d}$? I can not seem to find anything online for such a metric on matrices. (more on this below)
Secondly, I believe this pseudometric on $G$ is defined as follows: if we denote the representation by $\pi$ and the Euclidean metric by $d_e$, then set $d_G(g,h)=d_e(\pi(g),\pi(h))$. According to Wikipedia, this defines a pseudometric on $G$ if $d_e$ defines a pseudometric on $\mathbb{R}^{d\times d}$.
Is this the correct reasoning? If so, why is $d_e$ a pseudometric on $\mathbb{R}^{d\times d}$?
Edit: Following the hint from @Travis I think I understand this now. We take the Frobenius inner product on $\mathbb{R}^{d\times d}$ and the Euclidean metric induced by this. Defining the metric $d_G$ on $G$ as above only gives us a pseudometric (it only gives us a metric if the representation is faithful).