Practice Algebraic And Geometric Multiplicity (32.4) - Basis of Eigenvectors

Practice: Algebraic and Geometric Multiplicity

Lesson 32.4Basis of Eigenvectors

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9 questions

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6 sample questions

Question 1.
Define Algebraic Multiplicity.
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Relate it to counting occurrences.
Question 2.
What does Geometric Multiplicity represent?
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Think about the space spanned by eigenvectors.
3Question 3.
What is algebraic multiplicity?
- Number of linearly independent eigenvectors
- Dimension of eigenspace
- Times eigenvalue appears in the characteristic polynomial
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Think about counting citations in a research paper.
4Question 4.
True or False: The geometric multiplicity can be greater than the algebraic multiplicity.
- True
- False
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Reflect on their definitions.
Question 5.
For a matrix A with eigenvalues and their respective multiplicities: λ1=2, AM=3, GM=2; λ2=0, AM=1, GM=1. Discuss the diagonalizability of A.
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Refer back to the conditions that define diagonalizability and the role of eigenvalue multiplicities.
Question 6.
Explain why a symmetric matrix has only real eigenvalues. How does this impact AM and GM?
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Consider how the properties of symmetrical matrices simplify eigenvalue calculations.

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