Find the spectral radius for each matrix in question 5 find

Question

Question 1) Find l_∞ and l₂ norms of the vectors.

x = (sin k, cos k, 2^k)^t for a fixed positive integer k

Question 2) Find l_∞ norm of the matrices.

a.	10	15
	0	1

Question 3) The following linear system Ax = b have x as the actual solution and x˜ as an approximate solution. Compute ||x-x˜||_∞ and ||Ax˜ - b||_∞.

a. 1/2x₁ + 1/2x₂ = 1/63,

    1/3x₁ + 1/4x₂ = 1/168,

    x = (1/7 - 1/6)^t,

    x˜ = (0.142, 0.166)^t.

Question 4) Compute the eigenvalues and associated eigenvectors of the following matrices.

a.	2	-1
	-1	2

Question 5) Find the complex eigenvalues and associated eigenvectors for following matrices.

a.	2	2
	-1	2

Question 6) Find the spectral radius for each matrix in question 5.

Question 7) Find the l₂ norm for the matrices in question 5.

Question 8) Find the first two iterations of the Jacobi method for the following linear system, using x⁽⁰⁾ = 0:

a. 3x₁ - x₂ + x₃ = 1,

    3x₁ + 6x₂ + 2x₃ = 0,

    3x₁ + 3x₂ + 7x₃ = 4.

Question 9) Give an example of the matrix that is convergent.

Answer 1

The given questions as shown below about to matrices Question 1) Find l8 and l2 norms of the vectors. x = (sin k, cos k, 2k)t for a fixed positive integer k Question 2) Discover l8 norm of the matrices. a. 10 15 0 1 Question 3) The subsequent linear system Ax = b have x as the actual solution and x˜ as an approximate solution. Calculate ||x-x˜||8 and ||Ax˜ - b||8. a. 1/2x1 + 1/2x2 = 1/63, 1/3x1 + 1/4x2 = 1/168, x = (1/7 - 1/6)t, x˜ = (0.142, 0.166)t.