29.2.2. Steps to Find Eigenvectors

Try these first

1. 1.

   What is the first step to find an eigenvector?

   Hint

   Think about the equation we formed earlier.

2. 2.

   Explain what it means to find the null space.

   Hint

   What are we trying to solve for in the matrix?

3. 3.

   What is the first step to finding eigenvectors once an eigenvalue has been identified?

   • Substituting eigenvalue into (A - λI)
   • Calculating the determinant
   • Creating a characteristic polynomial

   Hint

   What do we do after finding eigenvalues to proceed to eigenvectors?

4. 4.

   True or False: Eigenvectors always correspond to unique eigenvalues.

   • True
   • False

   Hint

   Consider the relationship between values and their corresponding eigenvectors.

5. 5.

   Given the matrix A = [3 0; 0 2], find all eigenvectors for each eigenvalue and explain what they signify about the transformation represented by A.

   Hint

   Calculate using the substitution method and analyze each step closely.

6. 6.

   Matrix B has eigenvalues λ = -1 and λ = 4. Derive the eigenvectors and analyze their implications for the transformation behavior regarding compression and stretching.

   Hint

   Use the eigenvalue-eigenvector relationship carefully to derive implications.