32.1. Eigenvectors and Eigenspaces

Try these first

1. 1.

   Define what an eigenvector is.

   Hint

   Think about how it relates to matrices and transformations.

2. 2.

   What does the eigenspace correspond to?

   Hint

   Remember this definition involves the concept of null spaces.

3. 3.

   What is an eigenvector?

   Hint

   Think of the relationship between the vector v and the transformation A.

4. 4.

   An eigenvalue can have multiple corresponding eigenvectors. True or False?

   Hint

   Recall the definition of eigenspaces.

5. 5.

   Given the matrix A = [[2, -1], [1, 1]], find its eigenvalues and eigenvectors. Discuss the implications of your results on the matrix's diagonalizability.

   Hint

   Focus on the steps to derive the characteristic polynomial and the eigenvectors from there.

6. 6.

   Explore a 3x3 matrix with one eigenvalue repeated three times but only one linearly independent eigenvector. How would you illustrate this with an example, and what does it signify?

   Hint

   Remember to check both the characteristic polynomial and null space.