# Intuition behind $(X^TX)^{-1}$ in closed form of w in Linear Regression

The closed form of w in Linear regression can be written as

$\hat{w}=(X^TX)^{-1}X^Ty$

How can we intuitively explain the role of $(X^TX)^{-1}$ in this equation?

• Could you elaborate on what you mean by "intuitively"? For instance, there is a wonderfully intuitive explanation in terms of inner-product spaces presented in Christensen's Plane Answers to Complex Questions, but not everybody will appreciate that approach. As another example, there's a geometric explanation in my answer at stats.stackexchange.com/a/62147/919, but not everybody views geometrical relations as "intuitive." – whuber Aug 29 '18 at 17:02