Practice Example (32.5) - Basis of Eigenvectors - Mathematics (Civil Engineering -1)

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6 sample questions

Question 1.
What is the characteristic equation of the matrix A = [[4, 1], [0, 4]]?
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Remember, I is the identity matrix.
Question 2.
What is an eigenvalue?
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Think about how eigenvalues relate to transformations.
3Question 3.
What does the characteristic equation tell us?
- Determines eigenvalues
- Finds determinants
- Calculates traces
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Focus on how the term 'characteristic' relates to eigenvalues.
4Question 4.
True or False: The eigenspace dimension is always equal to the algebraic multiplicity of the eigenvalue.
- True
- False
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Review the definitions of AM and GM.
Question 5.
Given a new matrix B = [[4, 2], [0, 4]], calculate its eigenvalues and explain if it is diagonalizable.
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Check the relationship between AM and GM.
Question 6.
Suppose a matrix has real eigenvalues and the calculation reveals three linearly independent eigenvectors. What can be said about its diagonalizability?
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Recall the definition of diagonalizability.

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