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13. Given a matrix X and a vector y of appropriate dimensions, define f(w) = ||y – Xw||² %3D a. If X is N x k, what are the possible dimensions of w, y? b. Show that f(w) = y'y + w'X'Xw – 2w'X'y. c. Show that the gradient and Hessian are given by: = 2X'Xw – 2X'y Vf(w) V²f(w) X'X

13. Given a matrix X and a vector y of appropriate dimensions, define f(w) = ||y – Xw||² %3D a. If X is N x k, what are the possible dimensions of w, y? b. Show that f(w) = y'y + w'X'Xw – 2w'X'y. c. Show that the gradient and Hessian are given by: = 2X'Xw – 2X'y Vf(w) V²f(w) X'X

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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plz provide answer with explaination of (a),(b),(c).
13. Given a matrix X and a vector y of appropriate dimensions, define f(w) = ||y – Xw|? a. If X is N x k, what are the possible dimensions of w, y? b. Show that ƒ(w) = y'y + w'X'Xw – 2w'X'y. c. Show that the gradient and Hessian are given by: 2X'Xw – 2X'y Vf(w) V² f(w) X'X d. If X is symmetric and positive definite, show that f(w) is minimized at w* (X'X)-'X'y. e. What is minw f(w)?
Transcribed Image Text:13. Given a matrix X and a vector y of appropriate dimensions, define f(w) = ||y – Xw|? a. If X is N x k, what are the possible dimensions of w, y? b. Show that ƒ(w) = y'y + w'X'Xw – 2w'X'y. c. Show that the gradient and Hessian are given by: 2X'Xw – 2X'y Vf(w) V² f(w) X'X d. If X is symmetric and positive definite, show that f(w) is minimized at w* (X'X)-'X'y. e. What is minw f(w)?
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