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Let λ ER and consider the matrix J = 1 0 X 0 0 00 1 0 0 λ 1 0 0 X € M4. (a) Show that A is the only eigenvalue of J. What is its algebraic multiplicity? (b) Find a basis for E(J, λ) and determine the geometric multiplicity of X. (c) Using your answers in (a) and (b) together with Theorem 3.32, explain why J is not diagonalizable.

Let λ ER and consider the matrix J = 1 0 X 0 0 00 1 0 0 λ 1 0 0 X € M4. (a) Show that A is the only eigenvalue of J. What is its algebraic multiplicity? (b) Find a basis for E(J, λ) and determine the geometric multiplicity of X. (c) Using your answers in (a) and (b) together with Theorem 3.32, explain why J is not diagonalizable.

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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Let λ ER and consider the matrix J — X 1 00 0 X 1 0 0 0 X 1 000X € M4. (a) Show that λ is the only eigenvalue of J. What is its algebraic multiplicity? (b) Find a basis for E(J, λ) and determine the geometric multiplicity of λ. (c) Using your answers in (a) and (b) together with Theorem 3.32, explain why J is not diagonalizable.
Transcribed Image Text:Let λ ER and consider the matrix J — X 1 00 0 X 1 0 0 0 X 1 000X € M4. (a) Show that λ is the only eigenvalue of J. What is its algebraic multiplicity? (b) Find a basis for E(J, λ) and determine the geometric multiplicity of λ. (c) Using your answers in (a) and (b) together with Theorem 3.32, explain why J is not diagonalizable.
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