What is the exact meaning of the entries of a correlation coefficients matrix?
I have spent time researching this, but could find only approximate interpretations which give me no good understanding of results. It's very clear to me what a correlation coefficient of 1, 0 or -1 mean, but I dislike the simplistic explanations I found for the numbers in between (e.g., 0-0.3 is scarce correlation, 0.3-0.7 is moderate, 0.7-1 is strong).
This gives no understanding about the difference between 0.99 and 0.85. As an engineer, I want to be able to calculate a threshold above which results fulfill requirements.
There is only one precise numerical interpretation I found in this page, but from the way it's stated I'm not sure if it's exact:
The value of r squared is typically taken as “the percent of variation in one variable explained by the other variable,” or “the percent of variation shared between the two variables.”
Is this correct? Are there other properties of the correlation coefficient one can use to analyse and explain the behaviour of a set of signals. Any mathematical insight related to this parameter would be appreciated.