Solved – Matrix multiplication to find correlation matrix

I think the author is simply handy-wavy there. I believe it is assumed that the columns of $A$ have norm 1 and mean 0.

$A^TA$ is a Gram matrix. Given you are using random variables to construct $A$, $A^TA$ is approximately proportional to the covariance matrix (scale by $n$ the number of variables). The correlation matrix is simply the scaled version of the covariance matrix. Clearly if your random variables in the columns of $A$ are already normalized to unit-norm then $A^TA$ does not need any further normalization and it is immediately a correlation matrix.

For example using MATLAB:

 % Set the seed and generate a random matrix  rng(123);          A = randn(10,3);  A'*A   % ans =  %  4.867589606955550  -3.004809945345502  -1.615373090426428  % -3.004809945345502   5.356131932084330  -0.457643222208441  % -1.615373090426428  -0.457643222208441   5.574303027408192.   % Normalized now all the variables to have unit norm and mean zeros   A = A - repmat(mean(A),10,1);  A(:,1) = A(:,1)./norm(A(:,1));  A(:,2) = A(:,2)./norm(A(:,2));  A(:,3) = A(:,3)./norm(A(:,3));  A'*A   % ans =  %  1.000000000000000  -0.568373671796724  -0.336996715690262  % -0.568373671796724   1.000000000000000  -0.052272607974969  % -0.336996715690262  -0.052272607974969   1.000000000000000 

If we wanted to make the columns of $A$ orthogonal too, we would orthonormalize $A$. In that case we would do a Gram-Schmidt process on $A$; a process relating to a lot of wonderful things eg. Givens rotations, Householder transformation, etc.)