Let $M=W_1 + d W_2$ and $M^*=W_1^* + d W_2^*$ with $W_1, W_2, W_1^*, W_2^* \in \mathbb{R}^{rxr}$. We define $\mu = \sum_{I=1}^{n}x_i$ with the $x_i$ are drawn from $\mathcal{N}(0,I)$.
It is written in the paper I am reading, that \begin{align} L(W_1, W_2) &= \frac{1}{2} \mathbb{E}(||M^L\mu - (M^*)^L\mu||) \\ &= \frac{1}{2} \mathbb{E}(\langle (M^L - (M^*)^L)\mu, (M^L - (M^*)^L)\mu \rangle) \\ &= \frac{1}{2} \mathbb{E}(\langle (M^L - (M^*)^L), (M^L - (M^*)^L)\mu \mu ^T \rangle) \\ &= \frac{n}{2} || M^L - (M^*)^L ||^2_F \end{align}
Here I struggle to understand the passage from the second to the third line and from the third to the last line as we first have a scalar product between two vectors and then between two matrices (line 2->3). I am guessing that the final norm is the Frobenius norm and that the n comes from the fact that $Tr(\mathbb{E}(\mu \mu^T)) = n$ since $\mathbb{E}(\mu \mu ^T) = Cov(\mu) = nI$.