32.8. Orthogonal Basis (for Symmetric Matrices)

Try these first

1. 1.

   What are symmetric matrices?

   Hint

   Look at the definition of symmetric matrices.

2. 2.

   What is an eigenvalue?

   Hint

   Think about the equation Av = λv.

3. 3.

   All eigenvalues of symmetric matrices are:

   • True
   • False

   Hint

   Recall the Spectral Theorem.

4. 4.

   Eigenvectors corresponding to distinct eigenvalues of symmetric matrices are:

   • Scalable
   • Orthogonal
   • Dependent

   Hint

   Think about the meaning of orthogonality.

5. 5.

   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.

6. 6.

   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.