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Question 2 (1 point) 8 -18 Find the eigenvalues and eigenvectors of the matrix A = 18] (The 3 -7 same as in the previous problem.) di = 2, V1 = [1] and 12 = -1, V2 = - [11] [1] 3 21 = 1, V1 = ܒܗ ܟܬ and 12 = -2, V2 = 2 x = 1, V1 = and 12 = -2, V2 = [11 11 x = -2, Vi and 12 = -3, V2 [1]
3. Find all the eigenvalues and corresponding eigenspaces for the matrix B = 4. Show that the matrix B = 0 1 is not diagonalizable. 0 4] Lo 5. Let 2, and 1, be two distinct eigenvalues of a matrix A (2, # 12). Assume V1, V2 are eigenvectors of A corresponding to 11 and 22 respectively. Prove that V1, V2 are linearly independent.
0 -2 - The matrix A -11 2 2 -1 has eigenvalues 5 X = 3, A2 = 2, 13 = 1 Find a basis B = {V1, V2, v3} for R3 consisting of eigenvectors of A. Give the corresponding eigenvalue for each eigenvector vi.
8. Find a symmetric 3 x 3 matrix with eigenvalues 11, 12 , and , 13 and corresponding orthogonal eigenvectors vi , V2 , and V3 1 11 = 1, 12 = 2, 13 = 3, vi -=[:)--[:)--[;)] 1
(1 point) Find the eigenvalues and eigenvectors for A = [14 11 8 -4 3 The eigenvalue a + bi = has an eigenvector The eigenvalue a - bi = has an eigenvector
Suppose that the matrix A A has the following eigenvalues and
eigenvectors:
(1 point) Suppose that the matrix A has the following eigenvalues and eigenvectors: 2 = 2i with v1 = 2 - 5i and - 12 = -2i with v2 = (2+1) 2 + 5i Write the general real solution for the linear system r' = Ar, in the following forms: A. In eigenvalue/eigenvector form: 0 4 0 t MODE = C1 sin(2t) cos(2) 5 2 4 0 0...
Find the eigenvalues and eigenvectors of the given matrix. 3 -1 A= 8 -3 Enter the eigenvalues in ascending order. If the eigenvalues are equal, both answers should be the same. 1 11 = -1 X1 = where x1 = (..) ( ) 1 12 = 1 X2 = where x2 =
4. Consider 3 linearly independent vectors V1, V2, V3 E R3 and 3 arbi- trary numbers dı, d2, d3 € R. (i) Show that there is a matrix A E M3(R), and only one, with eigenvalues dı, d2, d3 and corresponding eigenvectors V1, V2, V3. (ii) Show that if {V1, V2, V3} is an orthonormal set of vectors. then the matrix A is symmetric.
Problem 8. (15 points) Find eigenvalues and eigenvectors of the follwing matrix 3 -2 0 A= -1 3-2 0 -1 3
Problem 8. (15 points) Find eigenvalues and eigenvectors of the follwing matrix 3 -2 0 A= -1 3-2 0 -1 3
Find a 3 × 3 matrix A with eigenvectors v1
= 1 2 3 with λ = 1, v2 = 0 −1 1 with λ = 2 and v3 =
1 1 1 with λ = 10. (Hint: A must be diagonalizable, A = P
DP −1 . Figure out P and D, then compute A directly.)
(6) Find a 3 x 3 matrix A with eigenvectors V1...