Practice Diagonalizability And Basis Of Eigenvectors (32.12)

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6 sample questions

Question 1.
Define diagonalizability in your own words.
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Consider the form A = PDP⁻¹.
Question 2.
What is an eigenvalue?
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Think about how it relates to eigenvectors in the equation Av = λv.
3Question 3.
What is the main condition for a matrix to be diagonalizable?
- It has at least one eigenvalue.
- It has n linearly independent eigenvectors.
- It has distinct eigenvalues.
- None of the above.
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Think about how many vectors are needed to span the space.
4Question 4.
True or False: All symmetric matrices are diagonalizable.
- True
- False
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Recall the properties of symmetric matrices.
Question 5.
Given a 3x3 matrix with eigenvalues 2 (multiplicity 2) and 3 (multiplicity 1), explain whether it can be diagonalized and justify your answer.
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Explore geometric vs algebraic multiplicity.
Question 6.
Demonstrate the process of diagonalization for a symmetric matrix with eigenvalues 4, 5, and 6. Calculate P and D, showing all steps.
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Use the characteristics of symmetric matrices.

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