Chapter 10Eigenvalues and Eigenvectors___________________________After reading this chapter, you will be able to1.Know the definition of eigenvalues and eigenvectors of a square matrix2.Find eigenvalues and eigenvectors of a square matrix3.Relate eigenvalues to the singularity of a square matrix4.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 x1and x2, and let us assume each spring has the same spring constant ‘k’. Then by applying Newton’s 2ndand 3rdlaw of motion to develop a force-balance for each mass we have)(1212121xxkkxdtxdm-+-=)(122222xxkdtxdm--=x1x2m1m2kk
Rewriting the equations, we have0)2(212121=+--xxkdtxdm0)(212222=--xxkdtxdm15,20,1021===kmmLet0)2(151021212=+--xxdtxd0)(152021222=--xxdtxdFrom vibration theory, the solutions can be of the formSinAxii=(290/-tϖwhereiA= amplitude of the vibration of mass ‘i’ϖ= frequency of vibration0/= phase shiftthen)0(222/--=tSinwAdtxdiiϖSubstituting ixand 22dtxdiin equations,-10A1ϖ2–15(-2A1+ A2) = 0-20A2ϖ2–15(A1- A2) = 0gives(-10ϖ2+ 30) A1–15A2= 0-15A1 + (-20ϖ2 + 15) A2= 0or(-ϖ2+ 3) A1–1.5A2= 0-0.75A1 + (-ϖ2 + 0.75) A2= 0In matrix form, these equations can be rewritten as
=+---+-0075.075.05.132122AAϖϖ=---0075.075.05.1321221AAAAϖLet ϖ2= λ--=75.075.05.13A=21AAX[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 vibrationλϖ=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 Aivery large and make the system unstable.What is the general definition of eigenvalues and eigenvectors of a square matrix?