The matrixhas eigenvalue X =-2 repeated three timesFind an-2-eigenvector for A:v=Give a 7-generalized-2-eigenvector.20Give a to-generalized -generalized-2-eigenvectorU=ATo be counford correct all three vectors must be entered and be consistent)3-54685

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Answered: The matrix has eigenvalue X = -2…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Assume that a 3 x 3 matrix A has eigenvalues 21 = 1,22=0.8, 23=0.6 With corresponding eigenvectors (₂), v₂ = (³) och (⁹). (a) Which 3-vectors can be written as a linear combination of the three eigenvectors? (b) What A¹v1, A2, Av3 respectively Av? (n is an arbitrary integer) (c) What about Anv when n→ ∞o? (It depends a little on the coefficients x1, x2, x3) (d) Now suppose that we have a 3×3 matrix with some eigenvalues 21, 22, 23 € R V₁ = V3 =
1 Mark as TRUE or FALSE. If they are true give an explanation, if they are false, give a coun- terexample. You really need to understand the reasons behind your statements. (a) Every square matrix has an eigenvalue over the real numbers. (b) Let A be an n x n matrix. A vector v can be an eigenvector with respect to two different eigenvalues. (c) If the vector v is an eingenvector for the eigenvalue = ; show that lim, m A"v = 0, where 0 is the zero vector 0,0, .. ., 0]. (d) Let A be an n x n invertible matrix. A vector v can be an eigenvector with respect to an eigenvalue c, then v is an eigenvector for A- with respect to the eigenvalue c 4 (e) 5 vectors in R are always linearly dependent. (f) 4 vectors in R are always linearly independent. 2. Decide whether the vectors (1, 2,0, -1], [2, 6, -3, -3], [3, 10, -6,-5] are linearly independent. What is the dimension of the span they generate? 3. Find a basis among the vectors (1, 2, 3], [2,6, 10], [0, –3, -6], [-1, -3, -5] (which are surely…
Q2) for the following Matrix ‘A°, Please determine the following: a. Eigenvalue of A b. Eigenvector of A, make your answer for a. and b, in the unit vector form.
-4 -2) -2) - Let A = -4 5. 2 , Vュー and v2 = Verify that v and v2 are %3D -2 eigenvectors of A. Then orthogonally diagonalize A.
Let M be a 2 × 2 matrix with eigenvalues ₁ = -1.2, A2 = 1 with corresponding eigenvectors Consider the difference equation with initial condition Xo = 5 3 V₁ = V2 = 2 xk+1 = Mxk Write the initial condition as a linear combination of the eigenvectors of M. That is, write x0 = c1V1 + C₂ V₂ = In general, xk= V1+ V2 ) * vi+ k ) v2 Specifically, x4 = For large k, xk≈ k
Let -4 -6 A = -1 -3 1 2 where eigenvalues for matrix A are A = 2, 1. Answer the following: (i) Find the geometric and algebraic multiplicity of each eigenvalue. (iii) Is A diagonalizable? If yes, diagonalize A. (ii) Compute A".
Determine if v is an eigenvector of the matrix A

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The matrix has eigenvalue X = -2 repeated three times. Find an -2-eigenvector for A V Give a -generalized-2-eigenvector. 19 Give a to-generalized -generalized-2-eigenvector 7. A off three vectors must be entered and be consistent) 3 4 -8 5 27