Matrix multiplication to find correlation matrix

Question

In this book on matrix factorizations, the author states the following, which I don't find to be true empirically. Is it true and under what conditions?

ADD: Trying to recreate the answer in R, what is the issue?

> D<-matrix(c(7,1,4,5,2,2,4,5,5,2,5,1,7,8,0,7),nrow=4,ncol=2,byrow=TRUE)
> D
     [,1] [,2]
[1,]    7    1
[2,]    4    5
[3,]    2    2
[4,]    4    5
> cor(D)
           [,1]       [,2]
[1,]  1.0000000 -0.3333333
[2,] -0.3333333  1.0000000
> 
> D[,1]<-D[,1]-mean(D[,1])
> D[,2]<-D[,2]-mean(D[,2])
> 
> D[,1]<-D[,1]/norm(as.matrix(D[,1]))
> D[,2]<-D[,2]/norm(as.matrix(D[,2]))
> D
            [,1]       [,2]
[1,]  0.50000000 -0.3214286
[2,] -0.04545455  0.2500000
[3,] -0.40909091 -0.1785714
[4,] -0.04545455  0.2500000
> t(D)%*%D
           [,1]       [,2]
[1,]  0.4214876 -0.1103896
[2,] -0.1103896  0.2602041

Answer 1

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.)