# Covariance matrix formula interpretation - what am I missing?

I'm reading a paper that outlines the calculation of a covariance matrix like the following:

$C=\displaystyle\sum^{N_b}_{i=1}\vec{x}_i\vec{x}_i^T$

What is the order of this matrix? My interpretation of the formula is something akin to dot products, but it's producing a vector not the expected matrix. Can someone explain where my interpretation is incorrect?

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A link to the paper would help, but to me it seems most plausible that the $x_i$'s are column vectors (say of length $n$) and hence $x_ix_i^T$ has $(j,k)$ entry $x_i^jx_i^k$, and all matrices are $n$-by-$n$. If you're working over the real numbers, this looks like a decomposition of a symmetric positive semidefinite matrix as a sum of positive rank one matrices. – Jonas Meyer Apr 22 '10 at 20:13
Yes, now that I have looked at the paper, it seems to be implicitly assumed that vectors are columns by default. Notice for example that the coefficients of $x_i$ with respect to an orthonormal basis $(e_j)_j$are given on page 3 by $a_j=x_i^Te_j$. – Jonas Meyer Apr 22 '10 at 20:24
Very helpful, thank you! – fbrereto Apr 22 '10 at 20:29