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(d) dim ker(A) = dim ker(B). Combine (b) and (c) to show that if A and B are similar matrices, then 2 (e) are similar as well. Show that if A and B are similar, then the matrices A – XIn and B – In (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 A as an eigenvalue of B. Use your answers to the previous parts to conclude that if A is an eigenvalue

(d) dim ker(A) = dim ker(B). Combine (b) and (c) to show that if A and B are similar matrices, then 2 (e) are similar as well. Show that if A and B are similar, then the matrices A – XIn and B – In (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 A as an eigenvalue of B. Use your answers to the previous parts to conclude that if A is an eigenvalue

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices
Problem 24EQ
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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
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
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