5. The four matricesSx,Sy,Szand I are defined bySx=(0 11 0),Sy=(0−ii0),Sz=(100−1),I=(1 00 1)wherei2=−1. Show thatS2x= I andSxSy=iSz, and obtain similar results by permutting x, y and z.Given that v is a vector with Cartesian components (vx,vy,vz), the matrix S(v) is defined asS(v) =vxSx+vySy+vzSzProve that, for general non-zero vectorsaandb,S(a)S(b) =a.bI+iS(a×b).Without further calculation, deduce that S(a) and S(b) commute if and only ifaandbare parallelvectors.

5. The four matricesSx,Sy,Szand I are defined bySx=(0 11 0),Sy=(0−ii0),Sz=(100−1),I=(1 00 1)wherei2=−1. Show thatS2x= I andSxSy=iSz, and obtain similar results by permutting x, y and z.Given that v is a vector with Cartesian components (vx,vy,vz), the matrix S(v) is defined asS(v) =vxSx+vySy+vzSzProve that, for general non-zero vectorsaandb,S(a)S(b) =a.bI+iS(a×b).Without further calculation, deduce that S(a) and S(b) commute if and only ifaandbare parallelvectors.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Linear Transformations

Section6.4: Transistion Matrices And Similarity

Problem 13E

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