Practice Algebraic Multiplicity (29.3.1) - Eigenvalues - Mathematics (Civil Engineering -1)

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6 sample questions

Question 1.
What is algebraic multiplicity?
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Consider how many times you can factor the polynomial.
Question 2.
Define geometric multiplicity.
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Think about how many directions correspond to that eigenvalue.
3Question 3.
What does algebraic multiplicity refer to?
- The number of eigenvectors for an eigenvalue
- The number of times an eigenvalue is a root of the characteristic polynomial
- The stability of the matrix
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Focus on root counting!
4Question 4.
True or False: Geometric multiplicity can be greater than algebraic multiplicity.
- True
- False
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Remember the inequalities we've discussed.
Question 5.
Consider a matrix with characteristic polynomial p(λ) = (λ - 3)³(λ - 2) = 0. Determine the algebraic and geometric multiplicities for each eigenvalue.
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Don't forget the concepts and check eigenspace dimensions!
Question 6.
If an eigenvalue's algebraic multiplicity is 5 and geometric is 3, what can you infer about the matrix properties in terms of stability?
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Link it back to applications in structural assessments.

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