30.4. Properties of Eigenvectors

Try these first

1. 1.

   What happens to an eigenvector if it is multiplied by a scalar?

   Hint

   Think about if it stretches or shrinks.

2. 2.

   If a matrix has 3 distinct eigenvalues, how many linearly independent eigenvectors will it have?

   Hint

   Remember the definition of distinct eigenvalues.

3. 3.

   What does it mean if eigenvectors are linearly independent?

   Hint

   Recall the definition of linear independence.

4. 4.

   True or False: All eigenvectors of symmetric matrices are complex.

   • True
   • False

   Hint

   Think about the properties of symmetric matrices.

5. 5.

   Given a 3x3 real symmetric matrix, prove that its eigenvectors corresponding to distinct eigenvalues are orthogonal.

   Hint

   Apply the eigenvalue equation and manipulate using the symmetry property.

6. 6.

   Consider a 4x4 matrix that has 4 distinct eigenvalues. How would you go about finding its eigenvectors? Discuss the method and implications for usage in scenarios such as structural analysis.

   Hint

   Remember to check for linear independence.