Because normalization reduces the influence of variables that have a higher variance. In raw data, if you have marathon times and heights for runners in a race, and you measure the times in minutes and the heights in inches, the influence on the principal components from the times will likely be much more.
Like all problems in statistics, it ought to depend on the specific task at hand. If there is some a priori reason to use the original scale (or a different re-weighting), it ought to be used. In general, PCA on correlation matrices is much preferred for exactly the reason you mention.
Like all problems in statistics, it ought to depend on the specific task at hand. If there is some a priori reason to use the original scale (or a different re-weighting), it ought to be used. In general, PCA on correlation matrices is much preferred for exactly the reason you mention.