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Suppose V is a vector space of finite dimension k and T: V→ V is a linear transformation. Suppose T has one single eigenvalue λER, and call n its geometric multiplicity. Which value(s) of n (if any) will result in [T] being similar to a diagonal matrix, for all bases a of V? Your "values" of n should be in terms of k. If no such value of n exists, explain why not. Clarification 1: In general there are two meanings of multiplicity that apply to eigenvalues. "Multiplicity of an eigenvalue" could mean "number of times (x - 2) appears in the characteristic polynomial x(x) ("Algebraic Multiplicity") or the dimension of the eigenspace E ("Geometric Multiplicity"). In this question, we are considering the \textbf{geometric multiplicity} n of λ. Clarification 2: Saying "There exists basis a of V for which [T] is diagonal" is equivalent to saying that "[T] is similar to a diagonal matrix for all bases & of V". Hint: Your solution can consider k>n, k = n, and k < n separately.

Suppose V is a vector space of finite dimension k and T: V→ V is a linear transformation. Suppose T has one single eigenvalue λER, and call n its geometric multiplicity. Which value(s) of n (if any) will result in [T] being similar to a diagonal matrix, for all bases a of V? Your "values" of n should be in terms of k. If no such value of n exists, explain why not. Clarification 1: In general there are two meanings of multiplicity that apply to eigenvalues. "Multiplicity of an eigenvalue" could mean "number of times (x - 2) appears in the characteristic polynomial x(x) ("Algebraic Multiplicity") or the dimension of the eigenspace E ("Geometric Multiplicity"). In this question, we are considering the \textbf{geometric multiplicity} n of λ. Clarification 2: Saying "There exists basis a of V for which [T] is diagonal" is equivalent to saying that "[T] is similar to a diagonal matrix for all bases & of V". Hint: Your solution can consider k>n, k = n, and k < n separately.

Advanced Engineering Mathematics
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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Suppose V is a vector space of finite dimension k and T : V → V is a linear transformation. Suppose T has one single eigenvalue λER, and call n its geometric multiplicity. Which value(s) of n (if any) will result in [T]% being similar to a diagonal matrix, for all bases a of V? Your "values" of n should be in terms of k. If no such value of n exists, explain why not. Clarification 1: In general there are two meanings of multiplicity that apply to eigenvalues. "Multiplicity of an eigenvalue" could mean "number of times (x - 2) appears in the characteristic polynomial x(x) ("Algebraic Multiplicity") or the dimension of the eigenspace E₁ ("Geometric Multiplicity"). In this question, we are considering the \textbf{geometric multiplicity} n of λ. Clarification 2: Saying "There exists basis a of V for which [T] is diagonal" is equivalent to saying that "[T] is similar to a diagonal matrix for all bases a of V". Hint: Your solution can consider k>n, k = n, and k < n separately.
Transcribed Image Text:Suppose V is a vector space of finite dimension k and T : V → V is a linear transformation. Suppose T has one single eigenvalue λER, and call n its geometric multiplicity. Which value(s) of n (if any) will result in [T]% being similar to a diagonal matrix, for all bases a of V? Your "values" of n should be in terms of k. If no such value of n exists, explain why not. Clarification 1: In general there are two meanings of multiplicity that apply to eigenvalues. "Multiplicity of an eigenvalue" could mean "number of times (x - 2) appears in the characteristic polynomial x(x) ("Algebraic Multiplicity") or the dimension of the eigenspace E₁ ("Geometric Multiplicity"). In this question, we are considering the \textbf{geometric multiplicity} n of λ. Clarification 2: Saying "There exists basis a of V for which [T] is diagonal" is equivalent to saying that "[T] is similar to a diagonal matrix for all bases a of V". Hint: Your solution can consider k>n, k = n, and k < n separately.
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