21.6.2. Finding Eigenvalues

Try these first

1. 1.

   Define what an eigenvalue is.

   Hint

   Think about the equation Av = λv.

2. 2.

   What is the first step in finding eigenvalues?

   Hint

   You will need the identity matrix.

3. 3.

   What is the characteristic equation used to find eigenvalues?

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

   Hint

   Check how we form the matrix with λ and the identity.

4. 4.

   Eigenvalues can be complex numbers.

   • True
   • False

   Hint

   Think about transformations in two-dimensional spaces.

5. 5.

   Given a matrix B = [[1, 2], [2, 1]], calculate its eigenvalues and explain their significance.

   Hint

   Remember to compute the determinant and find the roots.

6. 6.

   Consider the matrix C = [[0, 2], [-1, 0]]. Find the eigenvalues and describe what they indicate about the system.

   Hint

   Use the characteristic equation to analyze the matrix.