Exercise 3.3 Machine Learning

Exercise 3.3 Consider the hat matrix H X(XTX) XT, where X is an N by d +1 matrix, and XTX is invertible. (a) Show that H is symmetric. (b) Show that H H for any positive integer K (e) IiIlis the identity matrix of size N, show that (1-H) -- IH for any positive integer K (d) Show that trace(H) d+ 1, where the trace is the sum of diagonal elements. [Hint: trace(AB) trace(BA) Exercise 3.3 Consider the hat matrix H X(XTX) XT, where X is an N by d +1 matrix, and XTX is invertible. (a) Show that H is symmetric. (b) Show that H H for any positive integer K (e) IiIlis the identity matrix of size N, show that (1-H) -- IH for any positive integer K (d) Show that trace(H) d+ 1, where the trace is the sum of diagonal elements. [Hint: trace(AB) trace(BA)