# Intuitive interpretation of the covariance matrix

I am trying to get a "feel" for the covariance matrix. I know that $$v_iCov(X)v_j$$ gives the covariance between $$X$$ projected along $$v_i$$ and $$v_j$$. I'm curious if there's a similar intuitive interpretation of $$Cov(X)v$$ for any vector $$v$$.

Edit:

$$X$$ is a vector of $$n$$ random variables and $$v_i, v_j, v \in R^n$$

• What are $v_i$, $v_j$ and $v$? Jan 26 at 22:42
• A good linear algebra textbook will provide helpful explanations of dual spaces, which is the underlying fundamental concept here.
– whuber
Jan 27 at 15:04
• "Covariance is the difference between the average product and the product of averages" Jan 28 at 15:12
• @whuber that's really interesting can you point to any source? I have learned about dual spaces but I don't know how they relate. Jan 31 at 6:19
• Mervyn Stone (1987), "Coordinate-Free Multivariable Statistics", Clarendon Press, Oxford. Michael Wichura (2006), "The Coordinate-Free Approach to Linear Models," Cambridge Univ. Press.
– whuber
Jan 31 at 15:35