30.1. Preliminaries

Try these first

1. 1.

   Define what an eigenvector is in your own words.

   Hint

   Think about the practical meaning of direction and scaling.

2. 2.

   What is the characteristic equation used for?

   Hint

   Consider how it relates to the determinant.

3. 3.

   What is an eigenvector?

   • A zero vector
   • A non-zero vector unchanged in direction
   • Any random vector

   Hint

   Think of stretching without changing direction.

4. 4.

   True or False: Eigenvalues can be complex numbers.

   • True
   • False

   Hint

   Consider matrices with specific characteristics.

5. 5.

   Find the eigenvalues and eigenvectors for matrix B = [[3, 1], [1, 3]]. Explain the physical meaning of your results.

   Hint

   Work through the characteristic equation step by step.

6. 6.

   Discuss how the properties of eigenvectors impact the stability of systems in civil engineering under dynamic loads.

   Hint

   Consider real-world structures and their failure mechanisms.