30.2. Eigenvalue Problem

Try these first

1. 1.

   Define what an eigenvector is.

   Hint

   Think about what happens to a vector when multiplied by a matrix.

2. 2.

   What does the characteristic equation help you find?

   Hint

   Recall the determinant condition.

3. 3.

   What must be true for a matrix to have a non-trivial solution for its eigenvectors?

   • det(A) ≠ 0
   • det(A) = 0
   • A is a diagonal matrix

   Hint

   Refer to the conditions for finding eigenvalues.

4. 4.

   Solving the equation Ax = λx leads to what condition for eigenvalues?

   • True
   • False

   Hint

   Think about the transformations involved.

5. 5.

   Consider the matrix C = [[2, -1], [1, 0]]. Find its eigenvalues and eigenvectors and interpret their significance in a physical system.

   Hint

   Start with the determinant to find eigenvalues, then solve for eigenvectors.

6. 6.

   Given a linear transformation matrix D that represents a vibrating system, find the eigenvalues and discuss how they can predict the system's behavior under oscillations.

   Hint

   Relate your result back to the physical interpretation in terms of energy states.