Practice Diagonalization And Basis Of Eigenvectors (32.7)

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6 sample questions

Question 1.
What is diagonalization?
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Think of transforming a matrix to make its operations simpler.
Question 2.
Define an eigenvector.
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Recall the relationship involving a scalar and a matrix.
3Question 3.
What is necessary for a matrix to be diagonalizable?
- n distinct eigenvalues
- n linearly independent eigenvectors
- symmetric matrix
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Think about the definitions of eigenvalues and eigenvectors.
4Question 4.
True or False: Diagonalization can simplify matrix calculations.
- True
- False
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Remember the purpose of diagonalization.
Question 5.
Here is a 2x2 matrix A: [[3, 1], [0, 2]]. Determine if it can be diagonalized and find the matrices P and D if it can.
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Remember the properties of eigenvalues and eigenvectors.
Question 6.
Consider the 2x2 matrix B = [[1, 1], [0, 1]]. Show that it cannot be diagonalized.
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Evaluate eigenvalues and their counting carefully.

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