30.8. Orthogonality of Eigenvectors

Try these first

1. 1.

   What is an eigenvector?

   Hint

   Think about scaling and direction.

2. 2.

   What does orthogonal mean regarding vectors?

   Hint

   Consider the dot product result.

3. 3.

   What does the orthogonality of eigenvectors guarantee in the context of a symmetric matrix?

   • They are parallel to each other.
   • Their dot product is zero.
   • They have the same length.

   Hint

   Think about the geometric interpretation of orthogonality.

4. 4.

   True or False: All eigenvalues have orthogonal eigenvectors.

   • True
   • False

   Hint

   Consider the conditions under which the property holds.

5. 5.

   Given a symmetric matrix A, find its eigenvalues and eigenvectors. Verify if they are orthogonal.

   Hint

   Utilize matrix operations and evaluate the results.

6. 6.

   Discuss how non-orthogonality of eigenvectors could affect a structural dynamic analysis and provide an example.

   Hint

   Reflect on structural health and analyze specific cases.