Is a chi-square test for independence appropriate on a contingency table where one category is the unsupervised learning cluster? I have a data set that has been partitioned into four clusters by executing a clustering algorithm that used principal components from a principal component analysis (PCA). I then make a contingency table where the two categories are the cluster it belongs to and a feature from the original data set that was not used in the PCA. The feature is called "letter" and has possible values of a, b, and c.

Is it appropriate to perform a chi-square test for independence on this contingency table? My reservation is that the clusters are based on the sample and are not fixed for each data point.
I have other "big picture" reservations as well. I'm rather new to data analysis, and I'm wondering if I'm taking an unsupervised learning technique too far, when I should be performing a supervised learning technique where I use my principal components to predict the "letter". 
 A: On its face there is nothing "wrong" with this approach. You are asking a question about the degree to which your newly identified clusters differ in their relative frequencies on a separate variable in your data set. 
Take the categorical nature of the data out of the problem and consider the same situation, but you wanted to know whether there was reason to suspect your newly found categories differed on something such as age, a continuous variable. Running an ANOVA would be a reasonable approach to addressing the question. Your situation is no different, except that the measured variable of interest happens to be categorical. 
UPDATE:
A comment below correctly points out a potentially relevant distinction between the strength of an association and the likelihood or probability that an association exists. In using the word "degree" in the opening paragraph, I may have unintentionally blurred the line between the two. A chi-square test provides you with the probability that differences in relative frequencies among your groups occurred due to chance. As mentioned below, to estimate how large such differences are (i.e., a strength of association question) other methods exist. 
