Let A be a real 2×2 matrix with a complex eigenvalue A = a -bi (b 0) and an associated eigenvector v in C². a. Show that A(Re v) = a Re v +b Im v and A(Im v) = - b Re v +a Im v. [Hint: Write v = Re v +i Im v, and compute Av.] а -b ] 1 b. Verify that if A = PCP¯', where P = and C = b Re v Im v then AP = PC. a ..... a. If 1 = a - bi is an eigenvalue of A, corresponding to eigenvector v, then Av =
Let A be a real 2×2 matrix with a complex eigenvalue A = a -bi (b 0) and an associated eigenvector v in C². a. Show that A(Re v) = a Re v +b Im v and A(Im v) = - b Re v +a Im v. [Hint: Write v = Re v +i Im v, and compute Av.] а -b ] 1 b. Verify that if A = PCP¯', where P = and C = b Re v Im v then AP = PC. a ..... a. If 1 = a - bi is an eigenvalue of A, corresponding to eigenvector v, then Av =
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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![Let A be a real 2x2 matrix with a complex eigenvalue 1 = a - bi (b 0) and an associated eigenvector v in C2.
a. Show that A(Re v) = a Re v +b Im v and A(Im v) = - b Re v +a Im v.
[Hint: Write v = Re v +i Im v, and compute Av.]
- b
a
and C =
1
b. Verify that if A = PCP¯', where P = Re v Im v
then AP = PC.
.....
a. If 1 = a - biis an eigenvalue of A, corresponding to eigenvector v, then Av =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f523362-59ac-43d1-96c6-e213db7e4e4e%2Ffc67ca42-19ac-4c2a-ae54-57677a564938%2F2zu5n9c_processed.png&w=3840&q=75)
Transcribed Image Text:Let A be a real 2x2 matrix with a complex eigenvalue 1 = a - bi (b 0) and an associated eigenvector v in C2.
a. Show that A(Re v) = a Re v +b Im v and A(Im v) = - b Re v +a Im v.
[Hint: Write v = Re v +i Im v, and compute Av.]
- b
a
and C =
1
b. Verify that if A = PCP¯', where P = Re v Im v
then AP = PC.
.....
a. If 1 = a - biis an eigenvalue of A, corresponding to eigenvector v, then Av =
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