Practice Diagonalization Criteria (33.3) - Diagonalization

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6 sample questions

Question 1.
What is an eigenvalue?
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Think about how eigenvalues relate to the actions on eigenvectors.
Question 2.
How do you determine if a matrix is diagonalizable?
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Remember what you learned about linear independence.
3Question 3.
A matrix is diagonalizable if it has:
- At least one eigenvalue
- n linearly independent eigenvectors
- Only distinct eigenvalues
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Focus on the number of eigenvectors needed.
4Question 4.
True or False: A matrix can be diagonalized if its eigenvalue has an algebraic multiplicity greater than its geometric multiplicity.
- True
- False
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Think about the definitions of both multiplicities.
Question 5.
Prove that the matrix [[2, 0], [0, 2]] is diagonalizable.
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Calculate the eigenvalues first and verify independence.
Question 6.
Given the matrix [[0, 1], [0, 0]], discuss whether it can be diagonalized and justify.
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Remember to check the eigenvectors corresponding to the eigenvalue.

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