28.12. Eigenvalues and Eigenvectors of Linear Transformations

Try these first

1. 1.

   Define an eigenvector in your own words.

   Hint

   Think about how a vector behaves during scaling.

2. 2.

   What is the relationship between eigenvalues and eigenvectors?

   Hint

   Consider the equation T(v) = λv.

3. 3.

   What is an eigenvalue?

   • A vector that defines direction
   • A scalar that represents scaling
   • The transformation itself

   Hint

   Think about what is left unchanged in the direction of an eigenvector.

4. 4.

   True or False: Eigenvectors can be the zero vector.

   • True
   • False

   Hint

   Remember the definition of eigenvectors and their importance.

5. 5.

   Consider a matrix A = [[3, 1], [0, 2]]. Calculate the eigenvalues and eigenvectors.

   Hint

   Start by plugging λ back into (A − λI).

6. 6.

   Explore a real-world scenario where an engineer must consider the eigenvalues of a structure to ensure stability. Explain the implications of the eigenvalues on design choices.

   Hint

   Think about what happens if the natural frequency matches an external force frequency.