31.4 - Diagonalization and Similarity
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Practice Questions
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Define what it means for a matrix to be diagonalizable.
💡 Hint: Think about what forms a matrix can take when diagonalized.
What is an eigenvector?
💡 Hint: Consider how these vectors behave during transformation.
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Interactive Quizzes
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What does it mean for a matrix to be diagonalizable?
💡 Hint: Focus on how D changes the representation of A.
True or False: A matrix with multiple identical eigenvalues cannot be diagonalized.
💡 Hint: Consider the importance of eigenvector independence.
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Challenge Problems
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Let A = [[2, 1], [1, 2]]. Find the eigenvalues and demonstrate whether A is diagonalizable.
💡 Hint: Calculate the characteristic polynomial to find the eigenvalues.
Consider a symmetric matrix A. Prove that it has real eigenvalues and therefore is diagonalizable.
💡 Hint: Refer to the properties of symmetric matrices and the spectral theorem.
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