Welcome to Engineering Mathematics 2

1. Welcome to Engineering Mathematics 2 We will cover 2topics today 1. Eigenvalues 2. Eigenvectors

2. Lecture Recap If det A ≠ 0 then only a trivial solution exists. Last lecture we looked at systems of equations inthe form However, if det A = 0 then we can prove that an infinite number of non-trivial solutions exist. We deduced that the equations have the trivial solution x = 0, but there may also exist non-trivial solutions. These solutions will often be in terms of an arbitrary parameter. i.e.

3. Eigenvalues Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, they arose in the study of quadratic forms and differential equations. Last lecture we saw that with any homogenous linear equations A.x = 0 The non- trivial solution only exists if det A = 0. Consider, The determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering. They arise in common applications such as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems Where A is an n x n matrix and λ is a scalar, we can rearrange to get Which has non-trivial solutions if

4. Eigenvalues Question Find the eigenvalues of Consider the equation This is known as the characteristic equation of A. Clearly, this relationship will only hold for certain values of λ. Hence, These are called eigenvalues. Usually we list the eigenvalues as λ1, λ2 etc.. Therefore

5. Eigenvalues Question Hence, Find the eigenvalues of Thus, real matrices can have complex eigenvalues which will always occur in pairs of complex conjugates.

6. Eigenvalues Question Find the eigenvalues of

7. Eigenvectors Recall that, Associated with each eigenvalue of A, there will be non-trivial solutions to this equation. These are called eigenvectors and are sometimes denoted ‘s’. Hence, we can write Eigenvectors are a special set of vectors associated with a matrix equation. They are sometimes also known as characteristic vectors, proper vectors, or latent vectors. We can find the solution to this matrix equation using Gaussian elimination.

8. Eigenvectors Example Thus, Find the eigenvectors of If From a previous question we saw that the eigenvalues are Then Hence, x1= y1= α Hence Thus i.e. For all non-zero values of α.

9. Eigenvectors We can see that the eigenvectors are always written in term of an arbitrary variable. If Then Typically, we express eigenvalues without the variable (α and β in this case). Instead, we know that multiples of eigenvectors are also non-trivial solutions. I.e. in our example Hence, If Then, x2= -3y2/2 = β Gives us our independent eigenvectors with α = 1 and β = 3 respectively. For all non-zero values of β.

10. Eigenvectors The row reduction method gives us Question Find the eigenvectors of In a previous question, we found the eigenvalues these are

11. Eigenvectors Using the same method with the other two eigenvalues we obtain, Hence, y1= z1= α By back substitution we can also deduce that x1= α. Hence,

12. Eigenvectors Question Find the eigenvectors and eigenvalues of Hence, λ1 = 0 & λ2 = 1 Therefore

13. Eigenvectors If, λ2 = 1 Thus, Hence, we have only two independent eigenvectors.

14. Eigenvectors Question Find the eigenvectors and eigenvalues of Taking λ1 = 2, Hence, x1= z1& y1= 0. Thus, Hence, λ1 = 2 & λ2 = 1

15. Eigenvectors Therefore y2can be any value. i.e. Taking λ2 = 1, y2= γ Thus, Hence, 2.x2= z2 & y2= 0 Let, x2= β therefore z2= 2.β i.e. If β = 1 & γ = 0 then If β = 0 & γ = 1 then However, y2= 0 is only one solution. We have the equation Which gives us 2 independent eigenvectors for one eigenvalue.

16. Exam questions Given that v1 = (1, 2) and v2 = (3, -5) are eigenvectors of the matrix A, the corresponding eigenvalues are a) 1/4 and 1/7 b) -1/4 and 1/7 c) 4 and 7 d) -4 and 7

17. Conclusion Essential reading for next lecture HELM Workbook 22.1 Eigenvector Basics HELM Workbook 22.2 Applications of Eigenvalues and Eigenvectors We have covered 2 topics today 1. Eigenvalues 2. Eigenvectors