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Eigenvalues and eigenvectors of rotation matrices

Physics 116 AFall 2019 Eigenvalues and eigenvectors of rotation matricesThese notes are a supplement to a previous class handout entitled, rotation Matricesin two, three and many dimensions. In these notes, we shall focus on the Eigenvalues andeigenvectors of proper and improper rotation matrices in two and three The Eigenvalues and eigenvectors of proper and improper rotation matricesin two dimensionsIn the previous class handout cited above, we showed that the most general properrotation matrix in two-dimensions is of the form,R( ) = cos sin sin cos ,where 0 <2 ,(1)which represents a proper counterclockwise rotation by an angle in thex the eigenvalue problem,R( )~v= ~v.

tation that led to eq. (22). Moreover, the other two eigenvalues are complex conjugates of each other, whose real part is equal to cosθ, which uniquely fixes the rotation angle in the convention where 0 ≤ θ ≤ π. Case 1 corresponds to the identity (i.e. no rotation) and Case 2 corresponds to a 180 rotation about the axis nˆ. In Case 2 ...

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Transcription of Eigenvalues and eigenvectors of rotation matrices

1 Physics 116 AFall 2019 Eigenvalues and eigenvectors of rotation matricesThese notes are a supplement to a previous class handout entitled, rotation Matricesin two, three and many dimensions. In these notes, we shall focus on the Eigenvalues andeigenvectors of proper and improper rotation matrices in two and three The Eigenvalues and eigenvectors of proper and improper rotation matricesin two dimensionsIn the previous class handout cited above, we showed that the most general properrotation matrix in two-dimensions is of the form,R( ) = cos sin sin cos ,where 0 <2 ,(1)which represents a proper counterclockwise rotation by an angle in thex the eigenvalue problem,R( )~v= ~v.

2 (2)SinceR( ) rotates the vector~vby an angle , we conclude that for 6= 0 (mod ), thereare no real eigenvectors ~vthat are solutions to eq. (2). This can be easily checked by anexplicit calculation as (R( ) I) = 0 = det cos sin sin cos = 0,(3)which yields the characteristic equation,(cos )2+ sin2 = 0.(4)This equation simplifies to 2 2 cos + 1 = 0,(5)which yields the Eigenvalues , = cos cos2 1 = cos isin =e i .(6)Thus, we ave confirmed that the Eigenvalues are not real if 6= 0 (mod ). For the specialcases ofR=IandR= I, corresponding to = 0 and , respectively, we obtain realeigenvalues as expected. In particular, the case of = corresponds to a two dimensioninversion~x ~x, which implies that the eigenvalue ofR( ) is doubly degenerate andequal to case of improper rotations in two dimensions is more the previousclass handout cited above, we noted that the most general improper rotation matrix intwo-dimensions is of the form,R( ) = cos sin sin cos ,where 0 <2 ,(7)which can be expressed as the product of a proper rotation and a reflection,R= cos sin sin cos 100 1.

3 (8)However, it is easy to show that the action ofR( ) is equivalent to a pure reflectionthrough a line that passes through the origin. This can be seen by considering theeigenvalue problem,R( )~v= ~v.(9)We can again determine the Eigenvalues ofR( ) by solving its characteristic equation,det(R( ) I) = 0 = det cos sin sin cos = 0,(10)which is equivalent to(cos )( cos ) sin2 = 0.(11)This equation simplifies to 2 1 = 0,(12)which yields the Eigenvalues , = interpretation of this result is immediate. The matrixR( ) when operating ona vector~vrepresents a reflection of that vector through a line of reflectionthat passesthrough the origin.

4 In the case of = 1 we haveR( )~v=~v, which means that~vis avector that lies parallel to the line of reflection (and is thus unaffected by the reflection).In the case of = 1 we haveR( )~v= ~v, which means that~vis a vector that isperpendicular to the line of reflection (and is thus is transformed,~v ~v, by thereflection).One can therefore determine the line of reflection by computing theeigenvector thatcorresponds to = 1, cos sin sin cos xy = xy .(13)If = 0 (mod 2 ), then any vector of the form (x0) is an eigenvector corresponding to theeigenvalue = 1. This implies that the line of reflection is thex-axis, which correspondsto the equationy= 0.

5 In general (for any value of ), the solution to eq. (13) isxcos +ysin =x ,(14)2or equivalently,x(1 cos ) ysin = 0.(15)It is convenient to use trigonometric identities to rewrite eq. (15) as2xsin2 12 2ysin 12 cos 12 = 0.(16)If 6= 0 (mod 2 ), then we can divide both sides of eq. (16) by sin 12 to obtain1xsin 12 ycos 12 = 0.(17)We recognize eq. (17) as an equation for a straight line that passesthrough the originwith a slope equal to tan 12 . Thus, we have demonstrated that the most general2 2 orthogonal matrix with determinant equal to 1 given byR( ) represents a purereflection through a straight line of slope tan 12 that passes through the , it is worth noting that sinceR( ) is both an orthogonal matrix,R( )R( )T=I,and a symmetric matrix,R( )T=R( ), it follows that R( ) 2=I,(18)

6 Which is property that must be satisfied by a reflection matrix since two consecutivereflections are equivalent to the identity operation when acting on The Eigenvalues and eigenvectors of proper rotation matrices in threedimensionsThe most general three-dimensional proper rotation matrix, which we henceforthdenote byR( n, ), can be specified by an axis of rotation pointing in the directionof the unit vector n, and a rotation angle . Conventionally, a positive rotation anglecorresponds to a counterclockwise rotation . The direction of theaxis is determined by theright hand rule. Namely, curl the fingers of your right hand aroundthe axis of rotation ,where your fingers point in the direction.

7 Then, your thumb points perpendicular tothe plane of rotation in the direction of n. All possible proper rotations correspond to0 and the unit vector npointing in any learn more about the properties of a general three-dimensional rotation , considerthe matrix representationR( n, ) with respect to the standard basisBs={i,j,k}. Wecan define a new coordinate system in which the unit vector npoints in the directionof the newz-axis; the corresponding new basis will be denoted byB . The matrixrepresentation of the rotation with respect toB is then given byR(k, ) cos sin 0sin cos 0001 ,(19)where the axis of rotation points in thez-direction ( , along the unit vectork).

8 1 Note that for = 0 (mod 2 ), eq. (17) reduces toy= 0 which is the equation for thex-axis, the formalism developed in the class handout,Vector coordinates, matrix ele-ments, changes of basis, and matrix diagonalization, there exists an invertible matrixPsuch thatR( n, ) =P R(k, )P 1,(20)whereR(k, ) is given by eq. (19). In Appendix A, we will determine an explicit formfor the matrixP. However, the mere existence of the matrixPin eq. (20) is sufficientto provide a simple algorithm for determining the rotation axis n(up to an overall sign)and the rotation angle that characterize a general three-dimensional rotation determine the rotation angle , we note that the properties of the trace imply thatTr(P RP 1) = Tr(P 1P R) = TrR, since one can cyclically permute the matrices withinthe trace without modifying its value.

9 Hence, it immediately follows from eq. (20) thatTrR( n, ) = TrR(k, ) = 2 cos + 1,(21)after taking the trace of eq. (19). By convention, 0 , which implies that sin , the rotation angle is uniquely determined by eq. (21) To identify n, we observethat any vector that is parallel to the axis of rotation is unaffectedby the rotation last statement can be expressed as an eigenvalue equation,R( n, ) n= n.(22)Thus, nis an eigenvector ofR( n, ) corresponding to the eigenvalue 1. In particular, theeigenvalue 1 is nondegenerate for any 6= 0, in which case ncan be determined up to anoverall sign by computing the Eigenvalues and the normalized eigenvectors ofR( n, ).

10 Asimple proof of this result is given in Appendix B. Here, we shall establish this assertionby noting that the Eigenvalues of any matrix are invariant with respect to a similaritytransformation. In light of eq. (20), it follows that the eigenvaluesofR( n, ) are identicalto the Eigenvalues ofR(k, ). The latter can be obtained from the characteristic equation,(1 ) (cos )2+ sin2 = 0,which simplifies to:(1 )( 2 2 cos + 1) = 0,after using sin2 + cos2 = 1. Using the results of eqs. (5) and (6), it follows that thethree Eigenvalues ofR(k, ) are given by, 1= 1, 2=ei , 3=e i ,for 0 .There are three distinct cases:Case 1: = 0 1= 2= 3= 1 ,R( n,0) =I,Case 2: = 1= 1, 2= 3= 1 ,R( n, ) ,Case 3:0< < 1= 1, 2=ei , 3=e i ,R( n, ) ,where the corresponding rotation matrix is indicated for each of the three 6= 0 the eigenvalue 1 is nondegenerate, as expected from the geometric interpre-tation that led to eq.


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