<p><a href="https://stats.stackexchange.com/questions/22501/is-there-an-intuitive-interpretation-of-ata-for-a-data-matrix-a/22520#22520">Is there an intuitive interpretation of $A^TA$ for a data matrix $A$?</a> should be definitively awarded as it provides short but clear and pretty detailed answer for the very basic and very important question.</p>

<p>[Bounty started 7 April 2016, and has since been awarded]</p>
