Practice Geometric Multiplicity (29.3.2) - Eigenvalues - Mathematics (Civil Engineering -1)

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6 sample questions

Question 1.
Define geometric multiplicity.
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Think about the vectors corresponding to that eigenvalue.
Question 2.
What is algebraic multiplicity?
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Look at the characteristic polynomial.
3Question 3.
What defines geometric multiplicity?
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Think about independent vectors corresponding to eigenvalues.
4Question 4.
True or False: The geometric multiplicity of an eigenvalue is always greater than its algebraic multiplicity.
- True
- False
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Check the definitions of both multiplicities.
Question 5.
A 3x3 matrix has an eigenvalue λ = 4 that appears twice in its characteristic polynomial. Determine the possible geometric multiplicities and discuss von implications on diagonalization.
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Think about the definitions of multiplicities in regards to independence.
Question 6.
Consider the matrix A = [[1, 1], [0, 1]]. Calculate the eigenvalues, their multiplicities and determine if A is diagonalizable.
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Calculate the characteristic polynomial and confirm the case.

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