32.8 - Orthogonal Basis (for Symmetric Matrices)
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Practice Questions
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What are symmetric matrices?
💡 Hint: Look at the definition of symmetric matrices.
What is an eigenvalue?
💡 Hint: Think about the equation Av = λv.
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Interactive Quizzes
Quick quizzes to reinforce your learning
All eigenvalues of symmetric matrices are:
💡 Hint: Recall the Spectral Theorem.
Eigenvectors corresponding to distinct eigenvalues of symmetric matrices are:
💡 Hint: Think about the meaning of orthogonality.
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Challenge Problems
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Given the symmetric matrix [[1, 2], [2, 1]], find its eigenvalues and eigenvectors. Discuss how the orthogonality of the eigenvectors aids in analysis.
💡 Hint: Use the characteristic polynomial to solve for eigenvalues.
Discuss the implications if the eigenvectors of a symmetric 3x3 matrix were not orthogonal. How would it impact structural analyses?
💡 Hint: Think about how overlapping modes would complicate the decoupling of systems.
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