32.3. Steps to Find Basis of Eigenvectors

Try these first

1. 1.

   What is the first step to find the basis of eigenvectors for a given matrix?

   Hint

   Think about how you set up the determinant.

2. 2.

   What does the eigenspace correspond to?

   Hint

   Consider what defines a set of vectors for a given λ.

3. 3.

   What do you solve to find eigenvalues for matrix A?

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

   Hint

   What condition makes the determinant zero?

4. 4.

   True or False: The eigenspace for an eigenvalue includes only the zero vector.

   • True
   • False

   Hint

   Think about how many vectors make up an eigenspace.

5. 5.

   Given the 2x2 matrix A = [[1, 2], [0, 1]], calculate its eigenvalues and corresponding eigenvectors, discussing the implications of your findings.

   Hint

   What form will the characteristic polynomial take?

6. 6.

   For the matrix B = [[4, 1], [0, 4]], derive the eigenvectors and explain the relevance of the result for a repetitive eigenvalue.

   Hint

   How does the multiplicity of eigenvalues affect the linear independence of eigenvectors?