21.6.3. Finding Eigenvectors

Try these first

1. 1.

   Define eigenvector.

   Hint

   Consider how transformations affect direction and size.

2. 2.

   What does the characteristic equation determine?

   Hint

   Think about how this relates to the matrix's properties.

3. 3.

   What does an eigenvector represent in matrix transformations?

   • It is always zero
   • It changes direction
   • It scales by a factor

   Hint

   Remember, they don’t rotate under transformation.

4. 4.

   The characteristic equation is used to find:

   • True
   • False

   Hint

   Think about the properties that define eigenvalues.

5. 5.

   Given the matrix A = [[2, 1], [1, 2]], find the eigenvalues and corresponding eigenvectors.

   Hint

   Start with determinants and remember to substitute for eigenvalues.

6. 6.

   In structural engineering, explain the role of eigenvectors in analyzing the stability of a cantilever beam.

   Hint

   Reflect on vibration and structural response under load.