# What is the difference between $\mathbf X\mathbf X^\prime$ and $(\mathbf X-\mu)(\mathbf X-\mu)^\prime$?

In $\mathbf X\mathbf X^\prime$, $\mathbf X$ is a matrix contains data points in column fashion, $\mathbf X^\prime$ is its transpose, this looks like a covariance matrix, but does not subtract mean, so what is the different meaning of subtract mean or not subtract it?

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If you subtract the mean you get the covariance matrix, sample covariance matrix to be precise if you do not take expectations. If you do not subtract the mean, you get only the part of covariance matrix, since as @PEV wrote, to get the covariance matrix you need to subtract the term $\mu\mu^T$.
The covariance matrix is defined as $$\Sigma = E \left[(\textbf{X}-E[\textbf{X}])(\textbf{X}-E[\textbf{X}])^{T} \right]$$
$$= E(\textbf{X} \textbf{X}^{T})- \mu \mu^{T}$$