Setup:

Clear["Global`*"];

#1: 2D Rotations:

Clear["Global`*"];

With Mathematica, plot the Re and Im part of z = x + I y in the complex plane .
                        Plot3D[Im[Sqrt[x + I y]], {x, -4, 4}, {y, -4, 4}]
Plot3D[Re[Sqrt[x + I y]], {x, -4, 4}, {y, -4, 4}]
Comment on your answer .

x rotation

rotx={{Cos[θ],-Sin[θ]},{+Sin[θ],Cos[θ]}};
rotx //MatrixForm

Eigenvalues[rotx] //TrigToExp

Eigenvectors[rotx]

Transpose[rotx].rotx  //Simplify//MatrixForm

Det[rotx] //Simplify

vec={x,y}

vec2=rotx.vec;
vec2 //MatrixForm

point=vec2 /.{θ->π/6,x->1,y->0}

arrow[θIn_,vec_]:=Arrow[{{0,0},vec2/.{θ->θIn,x->vec[[1]],y->vec[[2]]}}]

arrow[π/6,{1,2}]

{Blue,arrow[π/6,{1,0}],Red,arrow[0,{1,0}]} //Graphics

{Blue,arrow[π/2,{1,0}],Red,arrow[0,{1,0}]} //Graphics

Eigenvalues[rotx]

Eigenvalues[rotx] //TrigToExp

#1: 3D Rotations:

With Mathematica, plot the Re and Im part of z = x + I y in the complex plane .
                        Plot3D[Im[Sqrt[x + I y]], {x, -4, 4}, {y, -4, 4}]
Plot3D[Re[Sqrt[x + I y]], {x, -4, 4}, {y, -4, 4}]
Comment on your answer .

Clear["Global`*"]

x rotation

vec={x,y,z};

rotx={{1,0,0},{0,Cos[θ],+Sin[θ]},{0,-Sin[θ],Cos[θ]}};
rotx //MatrixForm

tmp=rotx.vec //MatrixForm

Eigenvalues[rotx] //TrigToExp

Series[#,{θ,0,1}]& /@ tmp

Series[#,{θ,0,1}]& /@ tmp //Normal

rx.vec //MatrixForm

Eigenvalues[rotx]

Eigenvalues[rotx] //TrigToExp

evec=Eigenvectors[rotx];
evec //MatrixForm

evec=Eigenvectors[rotx];
evec //MatrixForm

evec = Normalize /@ evec;
evec //MatrixForm

ConjugateTranspose[evec].evec //MatrixForm

#1: Find axis and rotation angle

Matrix mat represents a finite rotation about a certain axis.
Find the axis and the angle of rotation.

Clear["Global`*"]

mat ={{(1/(Sqrt[2])),(1/(Sqrt[2])),0},{0,0,1},{(1/(Sqrt[2])),-(1/(Sqrt[2])),0}};
mat //MatrixForm

eq= 1+ 2 Cos[θ] == (1/(Sqrt[2]))

Solve[eq,{θ}] //N

Eigenvalues[mat]  //N

eq=Eigenvalues[mat]  [[1]]  //N

Solve[Cos[θ]==0.1464466094067262,θ]

Solve[Sin[θ]==-0.9892185757421227`,θ]

Abs[Eigenvalues[mat][[1]]]

NSolve[eq,{θ}]

evec=Eigenvectors[mat]

evec[[3]]

mat.evec[[3]] //Simplify

Transpose[mat].mat //MatrixForm

Det[mat]

#1: 3D Rotations:

xxx

Clear["Global`*"]

x rotation

rotx={{1,0,0},{0,Cos[θx],+Sin[θx]},{0,-Sin[θx],Cos[θx]}};
rotx //MatrixForm

roty={{Cos[θy],0,+Sin[θy]},{0,1,0},{-Sin[θy],0,Cos[θy]}};
roty //MatrixForm

rotz={{Cos[θz],+Sin[θz],0},{-Sin[θz],Cos[θz],0},{0,0,1}};
rotz //MatrixForm

vec={x,y,z};

tmp=rotx.vec //MatrixForm

tmpxy=rotx.roty.vec;
tmpxy //MatrixForm

tmpyx=roty.rotx.vec ;
tmpyx//MatrixForm

diff=tmpxy-tmpyx;
diff //MatrixForm

Series[f[x],{x,0,5}]

Series[f[x],{x,0,5}] //Normal

Series[diff,{θx,0,1},{θy,0,1}] //Normal //MatrixForm

Series[tmpxy,{θx,0,1},{θy,0,1}] //Normal  //Expand//MatrixForm

?Series

#1: Pauli Matrices

xxx

Clear["Global`*"]

x rotation

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

MatrixForm /@{s1,s2,s3}

fun[x_] = (x^2)

(#^2)&  /@ d

(#^2)& /@  {a,b,c}

fun[abc]

MatrixForm /@ {s1.s1,s2.s2,s3.s3}

MatrixForm /@ ((#.#)& /@  {s1,s2,s3})

comm[a_,b_]:= (a.b - b.a)

comm[s1,s2] //MatrixForm

LeviCivitaTensor[3][[1,2,3]]

2 I s3 LeviCivitaTensor[3][[1,2,3]]  //MatrixForm

comm[s1,s1]

comm[ s2,s3] == 2 I s1 LeviCivitaTensor[3][[2,3,1]]

comm[ s3,s1] == 2 I s2 LeviCivitaTensor[3][[3,1,2]]

comm[ s3,s2] == 2 I s1 LeviCivitaTensor[3][[3,2,1]]

LeviCivitaTensor[3][[1]] //MatrixForm

MatrixForm /@ Table[ LeviCivitaTensor[2][[i]],{i,1,2}]

MatrixForm /@ Table[ LeviCivitaTensor[3][[i]],{i,1,3}]

MatrixForm /@ Table[ LeviCivitaTensor[4][[i]],{i,1,4}]

#1: Infinitesimal Rotations:

xxx

Clear["Global`*"]

x rotation

rotx={{1,0,0},{0,Cos[θx],+Sin[θx]},{0,-Sin[θx],Cos[θx]}};
rotx //MatrixForm

roty={{Cos[θy],0,+Sin[θy]},{0,1,0},{-Sin[θy],0,Cos[θy]}};
roty //MatrixForm

rotz={{Cos[θz],+Sin[θz],0},{-Sin[θz],Cos[θz],0},{0,0,1}};
rotz //MatrixForm

rotx= Series[rotx,{θx,0,1}]  //Normal;
rotx//MatrixForm

roty=Series[roty,{θy,0,1}]  //Normal;
roty//MatrixForm

rotz= Series[rotz,{θz,0,1}]  //Normal;
rotz//MatrixForm

vec={x,y,z};

tmp=rotx.vec //MatrixForm

tmpxy=rotx.roty.vec;
tmpxy //MatrixForm

tmpyx=roty.rotx.vec ;
tmpyx//MatrixForm

diff=tmpxy-tmpyx;
diff //MatrixForm

#1: Infinitesimal Rotations: X,Y,Z

xxx

Clear["Global`*"]

x rotation

drotx={{0,0,0},{0,0,-1},{0,1,0}};
droty={{0,0,1},{0,0,0},{-1,0,0}};
drotz={{0,-1,0},{1,0,0},{0,0,0}};
MatrixForm /@{drotx,droty,drotz}

rotx={{1,0,0},{0,Cos[θx],+Sin[θx]},{0,-Sin[θx],Cos[θx]}};
rotx //MatrixForm

Series[rotx,{θx,0,1}] //Normal //MatrixForm

θx drotx  //MatrixForm

comm[a_,b_]:= (a.b - b.a)

comm[drotx,drotx] //MatrixForm

comm[drotx,droty] //MatrixForm

comm[droty,drotz] //MatrixForm

x

Parity: