4. Let x =20-10-and A =11116-21-1011 2 212(a) Show that is an eigenvector of A and find the associated eigenvalue.(b) Show that 3 is an eigenvalue of A and find an associated eigenvector.

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Answered: 4. Let a 2 0 -1 0 and A -1 1 1 2 – 1 1…

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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Consider the matrix 2 2 A = 4 2 2 (a) Compute the eigenvalues and eigenvectors of A. (b) Find rank one matrices B and C such that A = B + C and BC 0, explaining why the matrices you found have these properties. (hint: use your answer to part (a))
Verify that A, is an eigenvalue of A and that x, is a coresponding eigenvector. A = 5, x, = (1, 0, 0) え2= 3, X2 = (1, 2, 0) d3 = 4, x3 = (-6, 1, 1) 5-1 7 A = 3 1 %3D 0 4 5-1 7 Ax = 0. 3 1 = 1,X1 = 5 04 5 -1 7 1. Ax 2= 3 1 2. = 3 2 三 0 4 5 -1 7 Ax, =0 31 =4 0 4 1 H O
Find the eigenvalues of A, given that A = and its eigenvectors are The eigenvalues are -9 1 8 4 -8 -5 8 12 -4 11 8-8-8 and -1 -1 -2 and
[1] eigenbasis for A and thus diagonalize A. Let A = Show that 2 and 4 are eigenvalues of A and find all corresponding eigenvectors. Find an
Suppose that is an eigenvector for both the matrix A and the matrix B, with corresponding eigenvalue X for A and corresponding eigenvalue μ for B. Show that is an eigenvector of A + B and AB. Proof: Since Av = A and B = μv: (i) (A + B) = and (ii) ABv= A. Au + Bu B. A(Bu) = A(μv) C. λύ + μύ D. (X +μ) E. λμῦ F. μ(Aΰ) = μ(λύ) Q.E.D.
a) Show that is an eigenvalue of A and find one eigenvector corresponding to this eigenvalue. = = }]. A b)Show that is an eigenvalue of A and find one eigenvector corresponding to this eigenvalue. [3 1 A = 1 1 14 2 - λ = -6 1 0 λ = 2
2. (1) Compute all eigenvalues and eigenvectors of the Cartan matrix 2 -1 C = -1 2 -1 -1 2 (2) Diagonalize C (i.e. find Q such that Q-'CQ is diagonal). (3) Is C positive definite?
1. (a) Verify that ₁ and 2 are eigenvectors of A and find their eigenvalues A₁ and A2 where 3 4- [8 A = [B] 0 -1 [B] ,0₁ = and U₂ = (b) What do you notice about the eigenvalues and the matrix A. Note: upper triangular (or lower triangular) matrices always work that way.
Verify that A is an eigenvalue of A and that x, is a corresponding eigenvector. 4 -2 = -11, x; = (1, 2, –1) 22 = -3, x, = (-2, 1 0) 23 = -3, x; = (3, 0, 1) A = 2 -7 6 2 4 -2 13 Ax- 2 -7 6. 2 -11 2 !! 2 4 -2 2 AX2 -2 -7 !! %3D !! 4 -2 AX3 2 -7 = 13x3 1
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4. Let a 2 0 -1 0 and A -1 1 1 2 – 1 1 6 -2 1 0 1 2 2 (a) Show that is an eigenvector of A and find the associated eigenvalue. (b) Show that 3 is an eigenvalue of A and find an associated eigenvector.