Verify that li is an eigenvalue of A and that x; is a corresponding eigenvector. A...
Verify that ; is an eigenvalue of A and that x; is a corresponding eigenvector. A = [3 0 ] LO-3 14 = 3, x1 = (1, 0) 12 = -3, X2 = (0, 1) AX1 = = 3 = 11X1 10 -3 1 0 **(4- = -1) --- --6-418- | | |--[:)--- Ax2 = = -3 = 12x2 -3
Verify that i; is an eigenvalue of A and that x; is a corresponding eigenvector. 5 -1 4 A = 0 3 1 21 = 5, X1 = (1, 0, 0) 12 = 3, X2 = (1, 2, 0) 13 = 4, X3 = (-3, 1, 1) 0 04 5 -1 4 1 1 Ax1 = 0 3 1 0 = 5 0 11 III = 11X1 0 04 0 0 5 -1 4 1 1 Ax2 0 3 1...
Verify that i; is an eigenvalue of A and that x; is a corresponding eigenvector. -1 4 -6 A = 4 5 -12 -2 - 4 3 11 = 13, X1 = (1, 2, -1) 12 = -3, X2 = (-2, 10) 13 = -3, X3 = (3, 0, 1) 1 AX1 5 -12 = 13 2 = 11X1 -1 Ax2 4 5 -12 -2 -4 3 22X2 -1 3 3 4 5 -6 - 12 Ах3 4 1: -3...
DETAILS LARLINALG87.1.006. Verify that 2, is an eigenvalue of A and that x, is a corresponding eigenvector. 5 -1 2 an = 5, xn = 1, 0, 0) A =10 3 11; Ax = 3,x; = 1, 2, 0) 0 0 4 .[ (1 ,1 ,1-) = ;x,4 = ܨܬ 1 AX, ܐܐ 5 -1 2 0 3 1 0 0 4 ܐܐ ]. 0 ܕܫܢ AX, - 5 -1 2 0 3 1 0 0 4 2 =_32 ܕX,2...
13. Use three iterations of the power method to estimate the largest eigenvalue and corresponding eigenvector of A-2 4 to help with the arithmetic. Compare your estimates to the true values. For full credit, you must show all of your work and report each of the intermediate estimates xi, x2,A1, ?2 as well as the final estimates x3 and 23 ,starting with xo and ending with x3. You may use Matlab 0
3 7. If A is a 3x3 matrix with eigenvector o corresponding to an 1-21 eigenvalue of 5 and 2 corresponding to an eigenvalue of 2, and v= 7 [10] 4 find Av. 6
A = A has a = 5 as an eigenvalue, with corresponding eigenvector and i = 8 as an eigenvalue, with corresponding eigenvector . Find the solution to the system * = }}yı – žy2 y = - 5471 + 34 y2 that satisfies the initial conditions yı(0) = 0 and y2(0) = 3. What is the value of yı(1)?
Problem 2. Eigenvalue and Eigenvector Consider the mass-spring system in Fig. P13.5. The frequencies for the mass vibrations can be determined by solving for the eigenvalues and by applying Mi + kx = 0, which yields m 0 07/31 (2k -k -k X1 (0 0 m2 0 {2}+{-k 2k -kX{X2} = {0} LO 0 m3] 1 iz) 1-k -k 2kJ (x3) lo Applying the guess x = xoeiat as a solution, we get the fol- lowing matrix: 52k - m102...
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue 1. Let c be a scalar. Show that A-ch has eigenvector v, and corresponding eigenvalue X-c. (b) (8 marks) Let A = (33) i. Find the eigenvalues of A. ii. For one of the eigenvalues you have found, calculate the corresponding eigenvector. iii. Make use of part (a) to determine an eigenvalue and a corresponding eigenvector 2 2 of 5 - 1
Show that any two eigenvectors of the symmetric matrix corresponding to distinct eigenvalues are orthogonal. -1 0-1 0-1 0 -107 Find the characteristic polynomial of A. far - 41 - Find the eigenvalues of A. (Enter your answers from smallest to largest.) (11, 12, 13) = Find the general form for every eigenvector corresponding to 11. (Uses as your parameter.) X1 = Find the general form for every eigenvector corresponding to 12. (Use t as your parameter.) x2 = (0.t,0)...