4.F.1 Suppose the eigenvalues of a 3 × 3 matrix A are 2, 3/5, and 2/5 with corresponding eigenvec- tors 1 2 -3 vi = v3 = 34 v2 -3 -3 For this problem you don't need to know A!! a) Describe A(v1 + 2v2 + 3v3) as a linear combination of {v1, v2, v3}. -2 b) Let ro = -5 |. Describe ro as a linear combination of {v1, v2, v3}. 3 c) Find the solution of the equation Ik+1 = Axk for the specified ro (e.g. find an explicit formula for æµ in terms of the eigenvectors v¡). d) As k → 00, describe the behavior of æk. (Answer both the question: does lim æ exist? What can you say about the line through æk, or the angle between x and any other vectors.) e) As k → -00, describe the behavior of æk. (Same parenthetical.)
4.F.1 Suppose the eigenvalues of a 3 × 3 matrix A are 2, 3/5, and 2/5 with corresponding eigenvec- tors 1 2 -3 vi = v3 = 34 v2 -3 -3 For this problem you don't need to know A!! a) Describe A(v1 + 2v2 + 3v3) as a linear combination of {v1, v2, v3}. -2 b) Let ro = -5 |. Describe ro as a linear combination of {v1, v2, v3}. 3 c) Find the solution of the equation Ik+1 = Axk for the specified ro (e.g. find an explicit formula for æµ in terms of the eigenvectors v¡). d) As k → 00, describe the behavior of æk. (Answer both the question: does lim æ exist? What can you say about the line through æk, or the angle between x and any other vectors.) e) As k → -00, describe the behavior of æk. (Same parenthetical.)
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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Transcribed Image Text:4.F.1 Suppose the eigenvalues of a 3 × 3 matrix A are 2, 3/5, and 2/5 with corresponding eigenvec-
tors
--E} --} --E}
1
2
vi =
v2 =
v3 =
34
For this problem you don't need to know A!!
a) Describe A(v1 + 2v2 + 3v3) as a linear combination of {v1, v2, V3}.
-2
b) Let æo =
-5 . Describe ro as a linear combination of {v1, v2, v3}.
3
c) Find the solution of the equation ¤k+1 = Axk for the specified xo (e.g. find an explicit
formula for æk in terms of the eigenvectors v;).
d) As k → 00, describe the behavior of æk. (Answer both the question: does lim x exist? What
can you say about the line through Tk, or the angle between rk and any other vectors.)
e) As k → -0, describe the behavior of ær. (Same parenthetical.)
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