# Relationship between Correlation Matrix and Covariance Matrix

Given a matrix $$A = (X_1,X_2,...,X_n)$$

How is it that $$A^TA$$ is the correlation matrix where $$\frac{1}{n}(A^TA)_{ij} = Corr(X_i,X_j)$$?

I am confused because $$\frac{1}{n}(A^TA)_{ij} = \frac{1}{n}\sum_{k=1}^nA_{ki}A_{kj}$$

and $$\frac{1}{n}\sum_{k=1}^nA_{ki}A_{kj} = E[X_iX_j]$$

but how does $$Corr(X_i,X_j) = E[X_iX_j]$$ ?

I know that $$Cov(X_i,X_j) = E[X_iX_j] - \mu_i\mu_j$$

but $$Corr(X_i,X_j) = \frac{Cov(X_i,X_j)}{(Var(X_i)Var(X_j))^{\frac{1}{2}}}$$

Wouldn't this imply the following?

$$Cov(X_i,X_j) + \mu_i\mu_j = \frac{Cov(X_i,X_j)}{(Var(X_i)Var(X_j))^{\frac{1}{2}}}$$

Unless I am just attempting to skip some serious algebra here, I am not sure what I'm missing here...

This is because $$$$C_{X_i X_j} = \frac{Cov(X_i,X_j)}{(Var(X_i)Var(X_j))^{\frac{1}{2}}}$$$$ is the correlation coefficient determined by dividing the covariance by the product of the variables standard deviations, while the correlation is $$$$Corr(X_i,X_j)= E[X_i X_j]$$$$ If $$X_i$$ and $$X_j$$ have zero mean, this is the same as the covariance which is defined as $$$$Cov(X_i,X_j)= E[(X_i-\mu_{X_i}) (X_j-\mu_{X_j})]$$$$ with $$\mu_{X_i}$$ and$$\mu_{X_j}$$ the mean of $${X_i}$$ and $${X_j}$$, respectively.