Now explain why we may choose eigenvectors V₁ for ₁ and v₂ for ₂ that each have unit length and are orthogonal. Finally, define P and D using the data you've conjured so far such that A = PDP ¹ where P is an orthogonal matrix and D is diagonal. Delete this red text.
Now explain why we may choose eigenvectors V₁ for ₁ and v₂ for ₂ that each have unit length and are orthogonal. Finally, define P and D using the data you've conjured so far such that A = PDP ¹ where P is an orthogonal matrix and D is diagonal. Delete this red text.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:Now explain why we may choose eigenvectors V₁ for X₁ and V₂ for X₂ that
each have unit length and are orthogonal. Finally, define P and D using the
data you've conjured so far such that A = PDP ¹ where P is an orthogonal
matrix and D is diagonal. Delete this red text.
![Solution. Let A be a symmetric 2 x 2 matrix. Then, A has the form
a
b
[81]
b d
A
=
If b = 0, then A is orthogonally diagonalizable because...
Replace ... above with the reason why A is orthogonally diagonalizable when
b = 0. (What can you take P and D to be in this case?) Delete this red text.
Now suppose b +0. The characteristic polynomial of A is
(a-X) (d-X) - b² = λ² − (a + d)λ + ad-b².
From this it follows that the discriminant is
(a+d)² - 4(ad-b²) = (a - d)² + 4b².
Since b = 0, this quantity is always positive and hence the characteristic
polynomial has two distinct real roots. Thus A has two distinct real eigenvalues,
X₁ and X₂.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3886e624-3ad7-4547-aeab-f36b3d1dd04a%2Fe2e27d7c-4ed6-4ba1-9d48-b5591237c68b%2F646oihm_processed.png&w=3840&q=75)
Transcribed Image Text:Solution. Let A be a symmetric 2 x 2 matrix. Then, A has the form
a
b
[81]
b d
A
=
If b = 0, then A is orthogonally diagonalizable because...
Replace ... above with the reason why A is orthogonally diagonalizable when
b = 0. (What can you take P and D to be in this case?) Delete this red text.
Now suppose b +0. The characteristic polynomial of A is
(a-X) (d-X) - b² = λ² − (a + d)λ + ad-b².
From this it follows that the discriminant is
(a+d)² - 4(ad-b²) = (a - d)² + 4b².
Since b = 0, this quantity is always positive and hence the characteristic
polynomial has two distinct real roots. Thus A has two distinct real eigenvalues,
X₁ and X₂.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
