Practice - Algebraic and Geometric Multiplicity
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Practice Questions
Test your understanding with targeted questions
Define Algebraic Multiplicity.
💡 Hint: Relate it to counting occurrences.
What does Geometric Multiplicity represent?
💡 Hint: Think about the space spanned by eigenvectors.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What is algebraic multiplicity?
💡 Hint: Think about counting citations in a research paper.
True or False: The geometric multiplicity can be greater than the algebraic multiplicity.
💡 Hint: Reflect on their definitions.
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Challenge Problems
Push your limits with advanced challenges
For a matrix A with eigenvalues and their respective multiplicities: λ1=2, AM=3, GM=2; λ2=0, AM=1, GM=1. Discuss the diagonalizability of A.
💡 Hint: Refer back to the conditions that define diagonalizability and the role of eigenvalue multiplicities.
Explain why a symmetric matrix has only real eigenvalues. How does this impact AM and GM?
💡 Hint: Consider how the properties of symmetrical matrices simplify eigenvalue calculations.
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