Problem 1289 (difficulty: 6/10)

Given \(\displaystyle p_1,\ldots,p_n\) in 3-space we are looking for the plane through the origin for which the sum of the squared distances from the points to the plane is minimal. Let \(\displaystyle v\) be the normal vector of this plane, where \(\displaystyle |v|=1\).

(a) Show that \(\displaystyle v\) is an eigenvector of the matrix \(\displaystyle \sum\limits_{i=1}^n p_ip_i^T\).

(b) What is the geometric meaning of the eigenvalue corresponding to the eigenvector \(\displaystyle v\)?

Give me another random problem!