Find the Norm [[0.4,1],[0.6,0]]

Problem

‖[0.4,1],[0.6,0]‖

Solution

1. Identify the type of norm to be calculated. In the absence of a specific subscript, the spectral norm (the (L_2) operator norm) is typically used for matrices, which is the square root of the largest eigenvalue of AT*A

2. Calculate the product AT*A by multiplying the transpose of the matrix by itself.

AT*A=[[0.4,0.6],[1,0]]*[[0.4,1],[0.6,0]]

AT*A=[[0.16+0.36,0.4],[0.4,1]]

AT*A=[[0.52,0.4],[0.4,1]]

3. Find the characteristic equation det(AT*A−λ*I)=0 to determine the eigenvalues.

(0.52−λ)*(1−λ)−(0.4)*(0.4)=0

λ2−1.52*λ+0.52−0.16=0

λ2−1.52*λ+0.36=0

4. Solve for λ using the quadratic formula λ=(−b±√(,b2−4*a*c))/(2*a)

λ=(1.52±√(,(−1.52)2−4*(1)*(0.36)))/2

λ=(1.52±√(,2.3104−1.44))/2

λ=(1.52±√(,0.8704))/2

λ=(1.52±0.93295...)/2

5. Select the largest eigenvalue (λ_max())

(λ_max())=(1.52+0.93295...)/2

(λ_max())≈1.22647...

6. Compute the norm by taking the square root of the largest eigenvalue.

∥A∥=√(,1.22647...)

∥A∥≈1.10746...

Final Answer

‖[0.4,1],[0.6,0]‖=√(,0.76+√(,0.2176))≈1.1075

Want more problems? Check here!