Use (a) to solve the system x - ( )x. Х. -8 7

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
Use (a) to solve the system
x = (: :) .
Х.
-8 7
NB: Real solutions are required.
Hint: Recall that if the roots of C(A) = 0 occur in complex conjugate parts, then any one of
the two eigenvalues yields two linearly independent solutions. The second will yield solutions
which are identical (up to a constant) to the solutions already found.
a) is as follows
Show from first principles, i.e., by using the definition of linear independence, that if
x+iy, y # 0 is an eigenvalue of a real matrix A with associated eigenvector v = u + iw, then
the two real solutions
Y(t) = et (u cos bt – w sin bt)
and
Z(t) = eat (u sin bt + w cos bt)
are linearly independent solutions of X = AX.
Transcribed Image Text:Use (a) to solve the system x = (: :) . Х. -8 7 NB: Real solutions are required. Hint: Recall that if the roots of C(A) = 0 occur in complex conjugate parts, then any one of the two eigenvalues yields two linearly independent solutions. The second will yield solutions which are identical (up to a constant) to the solutions already found. a) is as follows Show from first principles, i.e., by using the definition of linear independence, that if x+iy, y # 0 is an eigenvalue of a real matrix A with associated eigenvector v = u + iw, then the two real solutions Y(t) = et (u cos bt – w sin bt) and Z(t) = eat (u sin bt + w cos bt) are linearly independent solutions of X = AX.
Expert Solution
steps

Step by step

Solved in 3 steps with 3 images

Blurred answer
Knowledge Booster
Linear Equations
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,