21.6.1. Definition

Try these first

1. 1.

   Define what an eigenvalue is in your own words.

   Hint

   Think about how vectors might change with a matrix.

2. 2.

   What does it mean for a vector to be an eigenvector?

   Hint

   Consider a vector that scales but does not rotate.

3. 3.

   What is the equation for finding eigenvalues?

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

   Hint

   Remember, we adjust the matrix A by subtracting λI.

4. 4.

   True or False: Eigenvectors corresponding to distinct eigenvalues are always linearly independent.

   • True
   • False

   Hint

   Think about the applicability of eigenvalues.

5. 5.

   Given the matrix M = [[3,1],[2,4]], find the eigenvalues and eigenvectors. Then explain how these concepts apply to real-world engineering problems.

   Hint

   Follow the steps through the characteristic equation.

6. 6.

   Calculate the eigenvalues for a 3x3 symmetric matrix and justify their properties.

   Hint

   Consider properties of symmetry!