# Solved – Matrix multiplication to find correlation matrix

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 ``
Contents

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.)

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