DISCOVERING STATISTICS USING SPSS (T H I R D E D I T I O N). Hope this will be helpful.

The correlation matrix is the standardized version of the covariance matrix. Analysing the correlation matrix is a useful default method because it takes the standardized form of the matrix; therefore, if variables have been measured using different scales this will not affect the analysis. Often you will want to analyse variables that use different measurement scales. Analysing the correlation matrix ensures that differences in measurement scales are accounted for. In addition, even variables measured using the same scale can have very different variances and this too creates problems for principal component analysis. Using the correlation matrix eliminates this problem also. There are statistical reasons for preferring to analyse the covariance matrix (the reason being that correlation coefficients are insensitive to variations in the dispersion of data whereas covariance is and so produces better-defined factor structures (Tinsley & Tinsley, 1987)) and generally the results will differ from analysis on the correlation matrix. However, the covariance matrix should be analysed only when your variables are commensurable.

I understand that one may adopt the covariance matrix when the variable scales are similar and the correlation matrix when variables are on different scales. What is the general position of people in the scientific world to this assertion?

DISCOVERING STATISTICS USING SPSS (T H I R D E D I T I O N). Hope this will be helpful.

The correlation matrix is the standardized version of the covariance matrix. Analysing the correlation matrix is a useful default method because it takes the standardized form of the matrix; therefore, if variables have been measured using different scales this will not affect the analysis. Often you will want to analyse variables that use different measurement scales. Analysing the correlation matrix ensures that differences in measurement scales are accounted for. In addition, even variables measured using the same scale can have very different variances and this too creates problems for principal component analysis. Using the correlation matrix eliminates this problem also. There are statistical reasons for preferring to analyse the covariance matrix (the reason being that correlation coefficients are insensitive to variations in the dispersion of data whereas covariance is and so produces better-defined factor structures (Tinsley & Tinsley, 1987)) and generally the results will differ from analysis on the correlation matrix. However, the covariance matrix should be analysed only when your variables are commensurable.

@Gevorg, thank you for the info. In fact, It really answered my apprehension, as per the info on nature of data set correlation is applicable to. Thanks alot

Perhaps you should also take into account another thing: unless you're analyzing truly continuous variables, using correlation matrixes instead of covariances might be useful as long as it allows you to choose correlation indexes others than Pearson's r.

I think you got this already, but here are my 2cents. I will give you a more practical answer. Depending on the nature of the data you could use either. If you have data that has different scales, temperature, pressure, etc etc .... you will want to use the correlation matrix since the variables are "standardized" by the their standard deviation so the total variance is equal to one. you are avoiding biasing the PCA by the magnitude of the change in the variables. In other words, if you use the covariance matrix your first PCs are going to be dominated by the variables with the greatest chance, which doesn't mean that it has the most "important" information, or the information you are looking for. If you are analyzing the responses from a set of sensors measuring the same property then the covariance matrix is appropriate. 

@Manuel, thanks for your wonderful contribution. I now have a clear understanding of the tools. In my current data analysis, it's important I adopt the correlation because of the different units of the variables. Thanks a bunch, Manuel. Tschauss

Hi

In practice, PCA using the covariance matrix is never used, and when it is, it makes litte sense if the original covariates are on different scales.

In un nutshell: I would always use a correlation matrix, which does not necessarily have to be Pearson's correlation, e.g. it can be Kendall's

https://arxiv.org/pdf/1305.6916.pdf

Hi There,

This is one is for Stefano. Stefano, that is a big statement. When you say "In practice" you probably mean: in your field. There are many, many situation where the correlation matrix is actually NOT APPROPRIATE. My background is in chemistry, so my go-to example would be spectroscopy data. In spectroscopy data you should never use the correlation matrix because you would be giving changes in low signal frequencies (mostly noise) equal standard deviation to actual signal... so, what you are doing is indeed lowering your signal-to-noise ratio. So, there is really not a "nutshell" approach. I agree that if you are not sure what to use it is safer to use the correlation matrix, but let's think an example where you are trying to add weight to come variables.... then when you use the correlation matrix you are effectively undoing the weighting.