7. Recall the following statement from class: If A and B are similar matrices, then A and B have the same eigenvalues, each occurring with the same algebraic and geometric multiplicities. In this problem, we will complete the proof of this statement by showing that if A is an eigen- value of A (and therefore also of B), then the geometric multiplicity of A as an eigenvalue of A must equal the geometric multiplicity of A as an eigenvalue of B. Suppose B = s-1AS, where A, B, and S are all n x n matrices. Show the (a) following two facts: (i) if v is in ker(B), then Sv is in ker(A). (ii) if w is in ker(A), then S-lw is in ker(B). Let v1, V2,..., Vm be a basis of ker(B). Show that the vectors Sv1, Sv2, Svm, which (by part (a)) are in ker(A), are linearly independent. (b) ... (Hint: Start by considering a nontrivial relation c1(Sv1)+c2(Sv2)+·· What happens if you multiply both sides of this relation by S-1?) ...+cm(Svm) = 0. (c) Show that the vectors Sv1, Sv2, ., Sv m span ker(A). .... (Hint: Let w be a vector in ker(A). What can you say about S-lw?) (d) dim ker(A) Combine (b) and (c) to show that if A and B are similar matrices, then = dim ker(B).
7. Recall the following statement from class: If A and B are similar matrices, then A and B have the same eigenvalues, each occurring with the same algebraic and geometric multiplicities. In this problem, we will complete the proof of this statement by showing that if A is an eigen- value of A (and therefore also of B), then the geometric multiplicity of A as an eigenvalue of A must equal the geometric multiplicity of A as an eigenvalue of B. Suppose B = s-1AS, where A, B, and S are all n x n matrices. Show the (a) following two facts: (i) if v is in ker(B), then Sv is in ker(A). (ii) if w is in ker(A), then S-lw is in ker(B). Let v1, V2,..., Vm be a basis of ker(B). Show that the vectors Sv1, Sv2, Svm, which (by part (a)) are in ker(A), are linearly independent. (b) ... (Hint: Start by considering a nontrivial relation c1(Sv1)+c2(Sv2)+·· What happens if you multiply both sides of this relation by S-1?) ...+cm(Svm) = 0. (c) Show that the vectors Sv1, Sv2, ., Sv m span ker(A). .... (Hint: Let w be a vector in ker(A). What can you say about S-lw?) (d) dim ker(A) Combine (b) and (c) to show that if A and B are similar matrices, then = dim ker(B).
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:7. Recall the following statement from class:
If A and B are similar matrices, then A and B have the same eigenvalues, each
occurring with the same algebraic and geometric multiplicities.
In this problem, we will complete the proof of this statement by showing that if A is an eigen-
value of A (and therefore also of B), then the geometric multiplicity of A as an eigenvalue
of A must equal the geometric multiplicity of A as an eigenvalue of B.
Suppose B = s-1AS, where A, B, and S are all n x n matrices. Show the
(a)
following two facts:
(i) if v is in ker(B), then Sv is in ker(A).
(ii) if w is in ker(A), then S-w is in ker(B).
Let v1, V2, . .., Vm be a basis of ker(B). Show that the vectors Sv1, Sv2,
Svm, which (by part (a)) are in ker(A), are linearly independent.
(b)
....
(Hint: Start by considering a nontrivial relation c1(Sv1)+c2(Sv2)+· ·+cm(Svm) = 0.
What happens if you multiply both sides of this relation by S-1?)
(c)
Show that the vectors Sv1, Sv2, ., Sv m span ker(A).
(Hint: Let w be a vector in ker(A). What can you say about S-lw?)
(d)
dim ker(A)
Combine (b) and (c) to show that if A and B are similar matrices, then
= dim ker(B).
(e)
Show that if A and B are similar, then the matrices A – XIn and B – XIn
are similar as well.
(Hint: Take another look at the proof given in class that similar matrices have the
same characteristic polynomial.)
(f)
of A (and therefore also of B), then the geometric multiplicity of A as an eigenvalue of
A must equal the geometric multiplicity of as an eigenvalue of B.
Use your answers to the previous parts to conclude that if is an eigenvalue
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 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,
