Practice Properties Of Eigenvectors (30.4) - Eigenvectors

Practise this section

9 questions

Flashcard drill
4 cards from this section. Quick, good the night before a test.
Exercise
6 questions · about 4 min · pass mark 70%. Hints available.
Sign up to start the exercise
Quiz
3 questions · about 2 min · pass mark 70%. Hints available.
Sign up to start the quiz

6 sample questions

Question 1.
What happens to an eigenvector if it is multiplied by a scalar?
Show hintHide hint
Think about if it stretches or shrinks.
Question 2.
If a matrix has 3 distinct eigenvalues, how many linearly independent eigenvectors will it have?
Show hintHide hint
Remember the definition of distinct eigenvalues.
3Question 3.
What does it mean if eigenvectors are linearly independent?
Show hintHide hint
Recall the definition of linear independence.
4Question 4.
True or False: All eigenvectors of symmetric matrices are complex.
- True
- False
Show hintHide hint
Think about the properties of symmetric matrices.
Question 5.
Given a 3x3 real symmetric matrix, prove that its eigenvectors corresponding to distinct eigenvalues are orthogonal.
Show hintHide hint
Apply the eigenvalue equation and manipulate using the symmetry property.
Question 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.
Show hintHide hint
Remember to check for linear independence.

5 more questions in this section. Enroll free to practise them all.