TITLE: What's the best way to think about the covariance matrix?
QUESTION [5 upvotes]: Let $X$ be a random vector with covariance matrix $\Sigma$.
People often describe $\Sigma$ in terms of its components:
$\Sigma_{ij}$ is the covariance of the $i$th and $j$th components of $X$.
But in linear algebra, thinking about a matrix in terms of its components is often discouraged.  It is often more enlightening to avoid thinking in terms of components.
So what is the "best" way to think about $\Sigma$, particularly for someone who likes linear algebra?
I know that $\Sigma = \mathbb E((X - \mu)(X - \mu)^T)$, where $\mu = \mathbb E(X)$.  But I think I am still missing something, because I'm not sure what to make of that formula.  Does this formula shed light on what $\Sigma$ really is and why we care about it?

REPLY [4 votes]: The best way to think about the covariance matrix: this is the matrix of the quadratic form
$$
\beta\in\Bbb R^d \to var \langle \beta, X\rangle
$$
Indeed,
\begin{align}
var \langle \beta, X\rangle &= var \sum_{i=1}^d \beta_i X_i \\
&=  \sum_{i=1}^d \sum_{j=1}^d \beta_i  \beta_j
cov( X_i ,X_j) \\
&=  \sum_{i=1}^d \sum_{j=1}^d \beta_i  \beta_j
\Sigma_{i,j} \\
&=  \beta^T\Sigma\beta \\
\end{align}