Can I run PCA (principal components analysis) if my Likert variables are measured differently? E.g. one Likert variable is in the range 0-6, another one is in the range 0-10 and yet another one is in the range 4-12. These are all risk variables. I want to keep them this way because the risk if you answered a 4 on one scale (first) would be equivalent to the risk if you answered 2 on another scale.
Usually, data is normalized before performing a PCA, as this post explains extensively. "Standardizing is usually done when the variables on which the PCA is performed are not measured on the same scale" and this quote seems to fit your problem very well.
If you don't normalize your data, variance will be dominated by the variables with larger ranges. In fact, not normalizing is like weighting your variables with their variances - that is, giving more importance to the variable with range 1-10 than the variable with range 1-6.
Furthermore, if you need to go back to your original scales, you can always denormalize any result.