Practice Orthogonal Basis (for Symmetric Matrices) (32.8)

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6 sample questions

Question 1.
What are symmetric matrices?
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Look at the definition of symmetric matrices.
Question 2.
What is an eigenvalue?
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Think about the equation Av = λv.
3Question 3.
All eigenvalues of symmetric matrices are:
- True
- False
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Recall the Spectral Theorem.
4Question 4.
Eigenvectors corresponding to distinct eigenvalues of symmetric matrices are:
- Scalable
- Orthogonal
- Dependent
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Think about the meaning of orthogonality.
Question 5.
Given the symmetric matrix [[1, 2], [2, 1]], find its eigenvalues and eigenvectors. Discuss how the orthogonality of the eigenvectors aids in analysis.
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Use the characteristic polynomial to solve for eigenvalues.
Question 6.
Discuss the implications if the eigenvectors of a symmetric 3x3 matrix were not orthogonal. How would it impact structural analyses?
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Think about how overlapping modes would complicate the decoupling of systems.

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