10. Suppose your friend is trying to diagonalize the matrix P, shown below. He calculates the eigenvalues of the matrix as 5 and 3, and finds that the eigenvectors of the matrix are fi [7, 2] and f₂ = [3, 1]7. From this information he designs the matrices T and D below, but when he (wisely) decides to validate his answer, he finds that PTDT-¹. Explain what has gone wrong. P=[] -9 42 -4 17 50 73 D-[88] - [23] D= 03
10. Suppose your friend is trying to diagonalize the matrix P, shown below. He calculates the eigenvalues of the matrix as 5 and 3, and finds that the eigenvectors of the matrix are fi [7, 2] and f₂ = [3, 1]7. From this information he designs the matrices T and D below, but when he (wisely) decides to validate his answer, he finds that PTDT-¹. Explain what has gone wrong. P=[] -9 42 -4 17 50 73 D-[88] - [23] D= 03
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
![10. Suppose your friend is trying to diagonalize the matrix P, shown below. He calculates the
eigenvalues of the matrix as 5 and 3, and finds that the eigenvectors of the matrix are f₁ =
[7, 2] and f2= [3, 1]7. From this information he designs the matrices T and D below, but
when he (wisely) decides to validate his answer, he finds that PTDT-¹. Explain what has
gone wrong.
P =
-9 42
-4
17
D=
50
03
T=
7 3
2 1
11. Suppose v = [2,3] and u = (3,5).
(a) Design a matrix A E R2x2 for which u and v are eigenvectors associated with the eigen-
values A = 4 and A= -2, respectively.
(b) Validate your matrix by demonstrating that Au = 4u and Av = -2v.
12. Suppose your friend Bucky calculates that A = 3 is an eigenvalue of the matrix H E R³×3.
Bucky writes down the matrix 31 - H (below) and asks you to find an associated eigenvector.
Sensing that Bucky lazy rather than lost, your first reaction is to say that you're busy enough](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F04a04416-13a1-4937-9d96-c9549c8907df%2F5a22368a-1a0b-41b1-bec0-677218b29f27%2Fxy9dxhw_processed.png&w=3840&q=75)
Transcribed Image Text:10. Suppose your friend is trying to diagonalize the matrix P, shown below. He calculates the
eigenvalues of the matrix as 5 and 3, and finds that the eigenvectors of the matrix are f₁ =
[7, 2] and f2= [3, 1]7. From this information he designs the matrices T and D below, but
when he (wisely) decides to validate his answer, he finds that PTDT-¹. Explain what has
gone wrong.
P =
-9 42
-4
17
D=
50
03
T=
7 3
2 1
11. Suppose v = [2,3] and u = (3,5).
(a) Design a matrix A E R2x2 for which u and v are eigenvectors associated with the eigen-
values A = 4 and A= -2, respectively.
(b) Validate your matrix by demonstrating that Au = 4u and Av = -2v.
12. Suppose your friend Bucky calculates that A = 3 is an eigenvalue of the matrix H E R³×3.
Bucky writes down the matrix 31 - H (below) and asks you to find an associated eigenvector.
Sensing that Bucky lazy rather than lost, your first reaction is to say that you're busy enough
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