Suppose we have n cases,  p variables
Then the data matrix is an n by p matrix with the ij-th cell being
  the value of the j-th variable for i-th case :

           x11  x12  x13   ........ x1p
           x21  x22  x23 ...........x2p
(  .................................. ...         )
           .....................................
          xn1  xn2 ..................xnp
 
 

If we substract the sample mean for each variable out, then the new data matrix
  would have column mean equal 0

For the rest of discussion,  we will work on this new data matrix.