NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN1 Lecture 15 Projection Least squares Projection matrix.

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Presentation transcript:

NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN1 Lecture 15 Projection Least squares Projection matrix

NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN2 Projection matrix C(P) = line through Rank(P)=1

NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN3 Properties of projection matrix : Why project? Because may have no solution Solve instead, where P is projection of b onto C(A) Plane of =column space of A is perpendicular to plane Project twice

NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN4 Find Key: is perpendicular to plane A What subspace of in ? Ans: e is in

NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN5 Projection matrix Case 1. A is not a square matrix so this equation is not true Case 2. A is a square matrix and invertible, means b is in C(A) than the projection is identity

NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN6 Check: Least squares Fitting by a line No solution Solve by projection Be continue…..

NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN7 Exercise: Let be a linear operator. For any Where p is the projection of z on the line find