Why data standardization is necessary in Prinicple Component Analysis (PCA) ?

Dear Guys,
Generally the PCA process is described as follows:
1: Get some data
[B]2: Standardize the data[/B]
3: Calculate the covariance matrix
4: Calculate the eigenvectors and eigenvalues of the covariance matrix
5: Choosing components and forming a feature vector
6: Deriving the new data set
My question is:
Why data Standardize is necessary? We know that by subtracting the mean from the data neither the variance nor the covariance change. So the covariance matrix stays unchanged. And the further steps are not affected. Thus, I guess Step 2 is useless. But unfortunately, many people and texts mentioned it. why is it like this?