Lecture 9: Eigensystems of Matrix EquationsLecture 9: Eigenvalues and Eigenvectors of a MatrixExample 9-1: Calculating Matrix Eigenvalues and EigenvectorsLecture 9: Symmetric, Skew-Symmetric, Orthogonal MatricesOrthogonal TransformationsExample 9-2: Coordinate Transformations to The Eigenbasis3.016 HomeJJ J I IIFull ScreenCloseQuitcW. Craig CarterSept. 26 2012Lecture 9: Eigensystems of Matrix EquationsReading:Kreyszig Sections: 8.1, 8.2, 8.3, 8.4Eigenvalues and Eigenvectors of a MatrixThe conditions for which general linear equationA~x =~b (9-1)has solutions for a given matrix A, fixed vector~b, and unknown vector ~x have been determined.The operation of a matrix on a vector—whether as a physical process, or as a geometric transformation, or just a generallinear equation—has also been discussed.Eigenvalues and eigenvectors are among the most important mathematical concepts with a very large number of applicationsin physics and engineering.An eigenvalue problem (associated with a matrix A) relates the operation of a matrix multiplication on a particular vector ~xto its multiplication by a particular scalar λ.A~x = λ~x (9-2)This equation indicates that the matrix operation can be replaced—or is equivalent to—a stretching or contraction of thevector: “A has some vector ~x for which its multiplication is simply a scalar multiplication operation by λ.” ~x is an eigenvectorof A and λ is ~x’s associated eigenvalue.3.016 HomeJJ J I IIFull ScreenCloseQuitcW. Craig CarterThe condition that Eq. 9-2 has solutions is that its associated homogeneous equation:(A − λI)~x =~0 (9-3)has a zero determinant:det(A − λI) = 0 (9-4)Eq. 9-4 is a polynomial equation in λ (the power of the polynomial is the same as the size of the square matrix).The eigenvalue-eigenvector system in Eq. 9-2 is solved by the following process:1. Solve the characteristic equation (Eq. 9-4) for each of its roots λi.2. Each root λiis used as an eigenvalue in Eq. 9-2 which is solved for its associated eigenvector ~xi3.016 HomeJJ J I IIFull ScreenCloseQuitcW. Craig CarterLecture 09 MathematicaR Example 1Calculating Matrix Eigenvalues and Eigenvectorsnotebook (non-evaluated) pdf (evaluated, color) pdf (evaluated, b&w) html (evaluated)The symbolic computation of eigenvalues and eigenvectors is demonstrated for simple 2 × 2 matrices. This example is illustrative—moreinteresting uses would be for larger matrices. In this example, a “cheat” is employed so that a matrix with “interesting” eigenvalues andeigenvectors is used as computation fodder.1mymatrix = 882 + Pi, -2 + Pi<, 8-2 + Pi, 2 + Pi<<;mymatrix êê MatrixFormK2 + p -2 + p-2 + p 2 + pOSolve the characteristic equation for the two eigenvalues:2Solve@Det@mymatrix - l IdentityMatrix@2DD ã 0, lDCompute the eigenvectors:3Eigenvectors@mymatrixD48evec1, evec2< = Eigenvectors@mymatrixDEigensystem will solve for eigenvalues and corresponding eigenvectors in one step:5Eigensystem@mymatrixD882 p, 4<, 881, 1<, 8-1, 1<<<Note the output format above: the first item in the list is a list of the two eigenvalues; the second item in the list is a list of the two corresponding eigenvectors. Thus, the eigenvector corresponding 2 p is (1,1).1: A “typical” 2 × 2 matrix mymatrix is defined for the calculations that follow. We will calculate itseigenvalues directly and with a built-in function.2: Its eigenvalues can be obtained by by using Solve for the characteristic equation Eq. 9-4 in termsof λ.3: And, its eigenvectors could be obtained by putting each eigenvalue back into Eq. 9-2 and then solving~x for each unique λ. However, this tedious procedure can also be performed with Eigenvectors4: Here, a matrix of eigenvectors is defined with named rows evec1 and evec2.5: Eigensystem generates the same results as Eigenvectors and Eigenvalues in one step.3.016 HomeJJ J I IIFull ScreenCloseQuitcW. Craig CarterThe matrix operation on a vector that returns a vector that is in the same direction is an eigensystem. A physical systemthat is associated can be interpreted in many different ways:geometrically The vectors ~x in Eq. 9-2 are the ones that are unchanged by the linear transformation on the vector.iteratively The vector ~x that is processed (either forward in time or iteratively) by A increases (or decreases if λ < 1) alongits direction.In fact, the eigensystem can be (and will be many times when they are) generalized to other interpretations and generalizedbeyond linear matrix systems.Here are some examples where eigenvalues arise. These examples generalize beyond matrix eigenvalues.• As an analogy that will become real later, consider the “harmonic oscillator” equation for a mass, m, vibrating with aspring-force, k, this is simply Newton’s equation:md2xdt2= kx (9-5)If we treat the second derivative as some linear operator, Lspringon the position x, then this looks like an eigenvalueequation:Lspringx =kmx (9-6)• Letting the positions xiform a vector ~x of a bunch of atoms of mass mi, the harmonic oscillator can be generalized toa bunch of atoms that are interacting as if they were attached to each other by springs:mid2xidt2=Xi’s near neighbors jkij(xi− xj) (9-7)3.016 HomeJJ J I IIFull ScreenCloseQuitcW. Craig CarterFor each position i, the j-terms can be added to each side, leaving and operator that looks like:Llattice=m1d2dt2−k120 −k14. . . 0−k21m2d2dt2−k230 . . . 0............ mid2dt2......mN−1d2dt2−kN−1 N0 0 . . . −kN N −1mNd2dt2(9-8)The operator Llatticehas diagonal entries that have the spring (second-derivative) operator and one off-diagonal entryfor each other atom that interacts with the atom associated with row i. The system of atoms can be written as:k−1Llattice~x = ~x (9-9)which is another eigenvalue equation and solutions are constrained to have unit eigenvalues—these are the ‘normalmodes.’• To make the above example more concrete, consider a system of three masses connected by springs.3.016 HomeJJ J I IIFull ScreenCloseQuitcW. Craig CarterFigure 9-5: Four masses connected by three springsThe equations of motion become:m1d2dt2−k12−k13−k14−k12m2d2dt20 0−k130 m2d2dt20−k140 0 m2d2dt2x1x2x3x4=k12+ k13+ k140 0 00 k120 00 0 k1300 0 0 k14x1x2x3x4(9-10)which can be written asL4×4~x = k~x