Practice Conditions For Diagonalizability (21.11.2) - Linear Algebra

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6 sample questions

Question 1.
Define a diagonalizable matrix.
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Remember the form of the diagonal matrix.
Question 2.
What must a matrix have to be diagonalizable?
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Think about the concepts of eigenvalues and eigenvectors.
3Question 3.
What is a necessary condition for a matrix to be diagonalizable?
- It has at least one zero eigenvalue
- It must have n linearly independent eigenvectors
- It must be an identity matrix
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Think about the definitions of eigenvectors.
4Question 4.
True or False: All matrices with distinct eigenvalues are diagonalizable.
- True
- False
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Recall how eigenvalues relate to diagonalizability.
Question 5.
Prove that the matrix A = [[1, 2], [0, 3]] is diagonalizable or not. Use computational methods to find eigenvalues and relations.
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Calculate eigenvalues using the characteristic polynomial.
Question 6.
Create a diagonalizable matrix of size 3x3 and show that it satisfies necessary diagonalizability conditions.
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Choose distinct eigenvalues and construct accordingly.

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