Rotation Matrices

Clear["Global`*"]

r1={{0,0,0},{0,0,-1},{0,1,0}};
r2={{0,0,1},{0,0,0},{-1,0,0}};
r3={{0,-1,0},{1,0,0},{0,0,0}};
MatrixForm /@ {r1,r2,r3}

unit3=DiagonalMatrix[{1,1,1}];

vec={x,y,z};
xVec={1,0,0};
yVec={0,1,0};
zVec={0,0,1};

Small Rotation about Z axis

rotz =unit3 + ϵ r3;
rotz //MatrixForm

rotz.vec

rotz.xVec

rotz.xVec - xVec

rotz.yVec

rotz.yVec - yVec

rotz.zVec

rotz.zVec - zVec

Small Rotation about X axis

rotz =unit3 + ϵ r3;
rotz //MatrixForm

rotz.vec

rotz.xVec

rotz.xVec - xVec

rotz.yVec

rotz.yVec - yVec

rotz.zVec

rotz.zVec - zVec

Small Rotation about Y axis

rotz =unit3 + ϵ r3;
rotz //MatrixForm

rotz.vec

rotz.xVec

rotz.xVec - xVec

rotz.yVec

rotz.yVec - yVec

rotz.zVec

rotz.zVec - zVec

Pauli Matrices

Clear["Global`*"]

s1={{0,1},{1,0}};
s2=I{{0,-1},{1,0}};
s3={{1,0},{0,-1}};
MatrixForm /@ {s1,s2,s3}

unit2=DiagonalMatrix[{1,1}];

vec2= x s1 + y s2 + z s3;
vec2 //MatrixForm

Small Rotation about Z axis

rotz= unit2 + I (ϵ/2)s3;
rotz //MatrixForm

rotzCT =rotz //Conjugate //Transpose

ϵ/: Conjugate[ϵ]=ϵ

?ϵ

rotzCT

matz= (rotz.vec2.rotzCT//Expand) /.{(ϵ^2)->0};
matz //MatrixForm

vec2 //MatrixForm

matz - vec2 //MatrixForm

Small Rotation about X axis

rotz= unit2 + I (ϵ/2)s3;
rotz //MatrixForm

rotzCT =rotz //Conjugate //Transpose

ϵ/: Conjugate[ϵ]=ϵ

?ϵ

rotzCT

matz= (rotz.vec2.rotzCT//Expand) /.{(ϵ^2)->0};
matz //MatrixForm

vec2 //MatrixForm

matz - vec2 //MatrixForm

Small Rotation about Y axis

rotz= unit2 + I (ϵ/2)s3;
rotz //MatrixForm

rotzCT =rotz //Conjugate //Transpose

ϵ/: Conjugate[ϵ]=ϵ

?ϵ

rotzCT

matz= (rotz.vec2.rotzCT//Expand) /.{(ϵ^2)->0};
matz //MatrixForm

vec2 //MatrixForm

matz - vec2 //MatrixForm