21.13.3 - Application
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Practice Questions
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What is the definition of a minimal polynomial?
💡 Hint: Think about the polynomial's role concerning the matrix.
True or False: The minimal polynomial must always be of higher degree than the characteristic polynomial.
💡 Hint: Consider their relationship.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What defines a minimal polynomial?
💡 Hint: Think about the polynomial's role in matrix equations.
True or False: The minimal polynomial only applies to square matrices.
💡 Hint: Recall the definition of a minimal polynomial.
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Challenge Problems
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You encounter a 3x3 matrix with a characteristic polynomial of (λ-1)(λ-2)². Determine the minimal polynomial and discuss its implications on stability.
💡 Hint: Reflect on the meaning of repeated roots in the context of Jordan forms.
Consider a matrix with eigenvalues 3, 4, and a repeated eigenvalue of 5. How would this affect its minimal polynomial and theorize on the potential behavior of the system modeled by this matrix?
💡 Hint: Think about how multiple eigenvalues affect system dynamics, particularly in control systems.
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