Practice Diagonalization And Similarity (31.4)

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6 sample questions

Question 1.
Define what it means for a matrix to be diagonalizable.
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Think about what forms a matrix can take when diagonalized.
Question 2.
What is an eigenvector?
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Consider how these vectors behave during transformation.
3Question 3.
What does it mean for a matrix to be diagonalizable?
- It can be transformed into a lower triangular matrix.
- It can be represented as D = P^{-1}AP.
- It has no eigenvalues.
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Focus on how D changes the representation of A.
4Question 4.
True or False: A matrix with multiple identical eigenvalues cannot be diagonalized.
- True
- False
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Consider the importance of eigenvector independence.
Question 5.
Let A = [[2, 1], [1, 2]]. Find the eigenvalues and demonstrate whether A is diagonalizable.
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Calculate the characteristic polynomial to find the eigenvalues.
Question 6.
Consider a symmetric matrix A. Prove that it has real eigenvalues and therefore is diagonalizable.
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Refer to the properties of symmetric matrices and the spectral theorem.

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