I'm reading these lecture notes. There's this equation:
For calculating the covariance matrix. I don't really understand how this results in a square covariance matrix. Cause for example the responsibility or $\tau$ will be a $N$ dimensional vector as in $N$ responsibilities for each data point $N$. Then $x_n-\mu_k$ will be a $N\times D$ where $D$ is the dimension of each point. So the covariance matrix should be a $D\times D$. But If you multiply these together then $\tau * (x_n-\mu_k) * (x_n-\mu_k)^T \rightarrow (1,N)*(N,D)*(D,N)=(1,N)$ by matrix multiplication so then when you sum over N you get a scalar rather than a $D\times D$. What am I getting wrong here?