Getting sense of and testing multiple correlations in two-block structure I am exploring a multidimensional data set. The variables can be mainly divided into two groups: A and B. I have calculated the correlation (Spearman's rho) matrix of this data set, and found that although the correlation between a variable from group A and one from group B is generally low, there are nonetheless some interesting patterns. For instance, variable A1 in group A has a (small yet) positive correlation with each variable in B, A2 has a (small yet) negative correlation with each variable in B, etc. 
Are these patterns potentially useful? Is there a way to formally measure them?
 A: Assuming you have ruled out influential points, as @Anony-Moussse has suggested...Whether these correlational patterns are potentially useful is entirely a domain-specific question--i.e., one that only you can answer, since you're the only one who knows what these variables consist of.  But if you want to further explore patterns in the correlations, you might be able to unearth more fundamental, underlying patterns through the use of either principal component analysis (if your data are objectively measured) or factor analysis (if they are not).  
A: @rolando2 has a nice answer; I just want to add a few points.  You can take the factor scores of A and those of B and correlate them.  This is a simplified, two-step version of canonical correlation (which in turn, is where this is done simultaneously--solving for weighted linear combinations of your original variables so as to maximize the correlations between the the sets).  However, I often think it's more useful to first get factors that make sense to you individually (i.e., without regard for how well they will correlate with factors in the other set), and then look at how they relate to each other.  In other words, if for example, you think of A1, A3, and A5 as being different measures of the same underlying construct, confirm this with a factor analysis, and then combine them into a single variable that does a better job of capturing what you really want to be talking about.  Also do this for everything else (e.g., maybe you think A2, A4 and A6 are related, and B1-B7, etc.).  Then see how your factors are related to each other, and whether these relationships are meaningful.  
