29.1. Definitions and Concepts

Try these first

1. 1.

   What is an eigenvalue?

   Hint

   Think about how a matrix can transform a vector.

2. 2.

   Define an eigenvector.

   Hint

   Look for vectors that keep their direction after transformation.

3. 3.

   What is the characteristic equation used to find eigenvalues?

   • det(A−λI)=0
   • det(A)=0
   • A−λI=0

   Hint

   Think about how we determine if a matrix is singular.

4. 4.

   True or False: An eigenvector can be the zero vector.

   • True
   • False

   Hint

   Remember the definition of an eigenvector.

5. 5.

   Given the matrix C = [[3, 0], [0, 4]], find the eigenvalues and describe their significance for a physical system.

   Hint

   Use the characteristic polynomial to determine the eigenvalues.

6. 6.

   Consider a dynamics problem in civil engineering where a building's stiffness matrix K has eigenvalues of 2 and 5. How would these eigenvalues affect the building's response?

   Hint

   Think about the implications of natural frequencies in structural dynamics.