spe_eigenvalues - Chapter 10 Eigenvalues and Eigenvectors After reading this chapter you will be able to 1 Know the definition of eigenvalues and

spe_eigenvalues - Chapter 10 Eigenvalues and Eigenvectors...

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Chapter 10 Eigenvalues and Eigenvectors ___________________________ After reading this chapter, you will be able to 1. Know the definition of eigenvalues and eigenvectors of a square matrix 2. Find eigenvalues and eigenvectors of a square matrix 3. Relate eigenvalues to the singularity of a square matrix 4. Use the power method to numerically find in magnitude the largest eigenvalue of a square matrix and the corresponding eigenvector. _________________________________ What does eigenvalue mean? The word eigenvalue comes from the German word “Eigenwert” where Eigen means “characteristic” and Wert means “value”. But what the word means is not on your mind! You want to know why do I need to learn about eigenvalues and eigenvectors. Once I give you an example of the application of eigenvalues and eigenvectors, you will want to know how to find these eigenvalues and eigenvectors. That is the motive of this chapter of the linear algebra primer. Can you give me a physical example application of eigenvalues and eigenvectors? Look at the spring-mass system as shown in the picture below. Assume each of the two mass-displacements to be denoted by x 1 and x 2 , and let us assume each spring has the same spring constant ‘k’. Then by applying Newton’s 2 nd and 3 rd law of motion to develop a force-balance for each mass we have ) ( 1 2 1 2 1 2 1 x x k kx dt x d m - + - = ) ( 1 2 2 2 2 2 x x k dt x d m - - = x 1 x 2 m 1 m 2 k k
Rewriting the equations, we have 0 ) 2 ( 2 1 2 1 2 1 = + - - x x k dt x d m 0 ) ( 2 1 2 2 2 2 = - - x x k dt x d m 15 , 20 , 10 2 1 = = = k m m Let 0 ) 2 ( 15 10 2 1 2 1 2 = + - - x x dt x d 0 ) ( 15 20 2 1 2 2 2 = - - x x dt x d From vibration theory, the solutions can be of the form Sin A x i i = ( 29 0 / - t ϖ where i A = amplitude of the vibration of mass ‘i’ ϖ = frequency of vibration 0 / = phase shift then ) 0 ( 2 2 2 / - - = t Sin w A dt x d i i ϖ Substituting i x and 2 2 dt x d i in equations, -10A 1 ϖ 2 –15(-2A 1 + A 2 ) = 0 -20A 2 ϖ 2 –15(A 1 - A 2 ) = 0 gives (-10 ϖ 2 + 30) A 1 –15A 2 = 0 -15A 1 + (-20 ϖ 2 + 15) A 2 = 0 or (- ϖ 2 + 3) A 1 –1.5A 2 = 0 -0.75A 1 + (- ϖ 2 + 0.75) A 2 = 0 In matrix form, these equations can be rewritten as
= + - - - + - 0 0 75 . 0 75 . 0 5 . 1 3 2 1 2 2 A A ϖ ϖ = - - - 0 0 75 . 0 75 . 0 5 . 1 3 2 1 2 2 1 A A A A ϖ Let ϖ 2 = λ [ ] - - = 75 . 0 75 . 0 5 . 1 3 A [ ] = 2 1 A A X [A] [X] - λ [X] = 0 [A] [X] = λ [X] In the above equation, ‘ λ ’ is the eigenvalue and [X] is the eigenvector corresponding to λ . As you can see that if we know ‘ λ ’ for the above example, we can calculate the natural frequency of the vibration λ ϖ = Why are they important? Because you do not want to have a forcing force on the spring- mass system close to this frequency as it would make the amplitude A i very large and make the system unstable. What is the general definition of eigenvalues and eigenvectors of a square matrix?

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