Practice Repeated Eigenvalues And Geometric Multiplicity (33.10) - Diagonalization

Practice: Repeated Eigenvalues and Geometric Multiplicity

Lesson 33.10Diagonalization

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9 questions

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3 cards from this section. Quick, good the night before a test.
Exercise
6 questions · about 4 min · pass mark 70%. Hints available.
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Quiz
3 questions · about 2 min · pass mark 70%. Hints available.
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6 sample questions

Question 1.
What is algebraic multiplicity?
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Think about how many times you can factor the polynomial.
Question 2.
If a matrix has a repeated eigenvalue, how can you check if it is diagonalizable?
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Check the linear independence of the eigenvectors.
3Question 3.
What defines the algebraic multiplicity of an eigenvalue?
- The number of linearly independent eigenvectors
- The number of times an eigenvalue appears in the characteristic polynomial
- The dimension of the eigenspace
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Think about the factors of the polynomial.
4Question 4.
True or False: A matrix with repeated eigenvalues is always diagonalizable.
- True
- False
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Consider an example to check this.
Question 5.
Create a 3x3 matrix with one eigenvalue of multiplicity 3. Show its algebraic and geometric multiplicity and check if it's diagonalizable.
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Start with the characteristic polynomial to find AM.
Question 6.
Construct a matrix that is diagonalizable with two distinct eigenvalues, and verify its multiplicities.
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Choose a simple diagonal matrix to illustrate diagonalizability.

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