Page 358 of Introduction to probability, second edition, by Blitzstein and Hwang, defines the sample covariance as
$$r = \dfrac{1}{n} \sum_{i = 1}^n (x_i - \bar{x})(y_i - \bar{y}),$$
where $\bar{x} = \dfrac{1}{n} \sum\limits_{i = 1}^n x_i$ and $\bar{y} = \dfrac{1}{n} \sum\limits_{i = 1}^n y_i$ are the sample means.
However, I have also seen the sample covariance (matrix) defined as
$$S = \dfrac{1}{n - 1} \sum\limits_{i = 1}^n (\mathbf{X}_i - \bar{\mathbf{X}})(\mathbf{X}_i - \bar{\mathbf{X}})^T.$$
Why is $n$ used in the denominator of the first definition, but $n - 1$ is used in the denominator of the second definition?
I would greatly appreciate it if people would please take the time to explain this.
EDIT:
My understanding is that this is a matter of convention, but why does this difference exist at all? Obviously, these do not produce the same value, so there is clearly a difference between one choice or the other.
