Covariance matrix of two matrices- how to calculate

Question

In maximum covariance analysis, to extract correlated columns, it is asked to calculate the covariance matrix. For two vectors, corvariance matrix is understood, COV(v1,v2) = v1*v2'

How do I calculate Covariance matrix of two matrices? I failed to find a definition to compute it. Thanks in advance for help.

Answer 1

Here is some code the will generate the covariance matrix using Matlab built in function of coding it up yourself.

Output:

meanx =

   -0.2320
    0.0400


CX =

   26.0522    5.8712
    5.8712   26.2326


meany =

   -0.1923
   -0.1358


CY =

   20.2713   -0.0902
   -0.0902   32.0136

---

Code:

clear all 
close all

sim = 1000;                    % number of order pairs to generate for
                               % X_1 and X_2
x = zeros(2, sim);             % pre-allocating x

% generating the order pairs with specified probabilities
for m = 1:sim
    u = rand(1,1);
    if u <= 0.25
        x(1, m) = -8;
        x(2, m) = 0;
    elseif u > 0.25 && u <= 0.5
        x(1, m) = 0;
        x(2, m) = -8;
    elseif u > 0.5 && u <= 0.75
        x(1, m) = 2;
        x(2, m) = 6;
    else
        x(1, m) = 6;
        x(2, m) = 2;
    end
end

% est mean of x
meanx = [sum(x(1, :))/sim, sum(x(2, :))/sim]'

% covariance code uncomment to see it gives the same the results 
% CX = zeros(2,2);               % pre-allocating est Cx
% xbar= zeros(2, sim);           % pre-allocating xbar
%
% % est covariance matrix of X by eq 9.46
% for m = 1:sim 
%     xbar(:, m) = x(:, m) - meanx;
%     CX = CX + xbar(:, m)*xbar(:, m)'/sim;
% end
% 
% CX
CX = cov(x.')                  % determines the covariance using Matlab's 
                               % built in cov 

% A is the eigenvector matrix of Cx given
A = [1/sqrt(2), -1/sqrt(2); 
     1/sqrt(2), 1/sqrt(2)];
y = zeros(2, sim);             % pre-allocating y

% transform random vector X by Y = AX
for m = 1:sim 
    y(:, m) = A*x(:, m);
end

% est mean of y
meany = [sum(y(1, :))/sim, sum(y(2, :))/sim]'

% covariance code uncomment to see it gives the same the results
% CY = zeros(2, 2);
% ybar = zeros(2, sim);          % pre-allocating ybar         
% 
% % est covariance matrix of Y by eq 9.46
% for m = 1:sim 
%     ybar(:, m) = y(:, m) - meany;
%     CY = CY + ybar(:, m)*ybar(:, m)'/sim;
% end
% 
% CY
CY = cov(y.')                  % determines the covariance using Matlab's 
                               % built in cov