Property $E[\mathbf{xx}^T] = \mathbf{\Sigma}-\mathbf{\mu\mu}^T$.

Question

I'm reading through Principles of Machine Learning by Murphy for the purpose of understanding Gaussian Processes. For now I just want to understand the multivariate Gaussian distribution but I'm having some issues. For some $N$ dimensional random vector $\mathbf{x}$, they define the covariance matrix to be $$ \mathbf{\Sigma} = \begin{pmatrix} V[X_1] & Cov[X_1,X_2] & \dots & Cov[X_1,X_N] \\ Cov[X_2,X_1] & V[X_2] & & \\ \vdots & & \ddots & \\ Cov[X_N,X_1] & \dots & & V[X_N] \end{pmatrix}. $$ This is fair. But they write that $E[\mathbf{xx}^T] = \mathbf{\Sigma}-\mathbf{\mu\mu}^T$. In my mind, $\mathbf{xx}^T$ is just going to be a scalar because you have an $N\times 1$ dimensional vector multiplied by a $1 \times N$ dimensional one. But what sense does this make? The covariance matrix isn't a scalar? What am I missing here?

Answer 1

You answered yourself. $X$ is $N\times 1$ dimensional. Then $X^T$ is $1\times N$ dimensional. From which $XX^T$ will be $N\times N$ dimensional, a square matrix of dimension $N$.

The scalar product is $X^TX$.