Practice Diagonalization Of A Matrix (29.5) - Eigenvalues

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6 sample questions

Question 1.
What does it mean for a matrix to be diagonalizable?
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Consider what forms the P and D matrices take.
Question 2.
Define eigenvector in your own words.
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Think about how vectors behave under matrix multiplication.
3Question 3.
What is a diagonalizable matrix?
- A matrix with all zero eigenvalues
- A matrix that can be expressed as A = PDP⁻¹
- A matrix that is only square
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Think about the definition provided earlier.
4Question 4.
True or False: The diagonal matrix D only contains elements on its diagonal.
- True
- False
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Remember the definition of a diagonal matrix.
Question 5.
Given a matrix A = [[4, 1], [1, 4]], find if it is diagonalizable, and if so, compute the diagonalized form.
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Start by finding the characteristic polynomial to determine eigenvalues.
Question 6.
If a system of differential equations can be expressed in matrix form as dx/dt = Ax, how would diagonalization of A assist in solving this system?
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Consider how each eigenvalue influences the shape of the solution curve.

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