33.10 - Repeated Eigenvalues and Geometric Multiplicity
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Practice Questions
Test your understanding with targeted questions
What is algebraic multiplicity?
💡 Hint: Think about how many times you can factor the polynomial.
If a matrix has a repeated eigenvalue, how can you check if it is diagonalizable?
💡 Hint: Check the linear independence of the eigenvectors.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What defines the algebraic multiplicity of an eigenvalue?
💡 Hint: Think about the factors of the polynomial.
True or False: A matrix with repeated eigenvalues is always diagonalizable.
💡 Hint: Consider an example to check this.
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Challenge Problems
Push your limits with advanced challenges
Create a 3x3 matrix with one eigenvalue of multiplicity 3. Show its algebraic and geometric multiplicity and check if it's diagonalizable.
💡 Hint: Start with the characteristic polynomial to find AM.
Construct a matrix that is diagonalizable with two distinct eigenvalues, and verify its multiplicities.
💡 Hint: Choose a simple diagonal matrix to illustrate diagonalizability.
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