To my perception, normalization of data dimension is done to bring them to same scale. Mean is subtracted first (for zero centering) and then divided by s.d. (normalization)..or we may normalize to some scale like -1 to 1 depending on the data. Its a data pre-processing step so that PCA gives faithful result.
One of the main property of PCA is that it reduces dimensionality that is if we take a MxN matrix , you will transform using PCA then we have the Eigen values ( unique solutions or simply roots of an equation )