Linear Operators 1.1 2004/06/07 11:09:42 GMT-5 2004/06/15 00:50:03.031 GMT-5 Richard G. Baraniuk richb@rice.edu Richard G. Baraniuk richb@rice.edu Harika Basana ilsai@rice.edu Jeffrey Silverman jsilv@rice.edu linear linearity nonlinearity Introduction to linear operators. We will focus on linear operators in 301. linear An operator O is linear if O 0 0 ().

Superposition: O x 1 x 2 O x 1 O x 2 . then .

Scaling: O γ x γ O x , where γ . then .

We can combine requirements 2 and 3 into: O γ 1 x 1 γ 2 x 2 γ 1 O x 1 γ 2 O x 2 is equivalent to .

Right Hand Side

Left Hand Side

Check for linearity/nonlinearity.

DC Offset

Squarer

2-point Smoother

3-point Smoother

Median Filter

Edge Dectector

Matrix Multiply

Fact 1 All matrix multiply operators are linear. y A x where x N , y N , and A is N N ().

Proof Verify the 3 conditions for y k 0 N 1 a k x k where a k are the columns of matrix A, A a 0 a 1 … a N - 1 . Let x 0 , then all x k 0 . This implies that y 0 . Show A x 1 x 2 A x 1 A x 2 . LHS k 0 N 1 a k x 1 k x 2 k k 0 N 1 a k x 1 k k 0 N 1 a k x 2 k A x 1 A x 2 RHS Show A γ x γ A x LHS k 0 N 1 a k γ x k γ k 0 N 1 a k x k γ A x RHS

Fact 2 Only matrix multiply operators are linear.

Upshot of Facts 1 and 2: The Study of linear operators in ℂ N is equivalent to the study N N matrices.