Rotation & Reflection Matrix

                         
  Angle A   degree   radian cos   sin   cos2A   sin2A    
  θ td1 tr1 ctr1 str1 c2tr1 s2tr1  
  φ pd1 pr1 cpr1 spr1 c2pr1 s2pr1  
  θ+φ tpd1 tpr1 ctpr1 stpr1 c2tpr1 s2tpr1  
  θ-φ tpd2 tpr2 ctpr2 stpr2 c2tpr2 s2tpr2  
  φ-θ ptd2 ptr2 cptr2 sptr2 c2ptr2 s2ptr2  
  2(θ+φ) tpd1a tpr1a ctpr1a stpr1a c2tpr1a s2tpr1a  
  2(θ-φ) tpd2a tpr2a ctpr2a stpr2a c2tpr2a s2tpr2a  
  2(φ-θ) ptd2a ptr2a cptr2a sptr2a c2ptr2a s2ptr2a  
  θ+ φ/2 tpd3 tpr3 ctpr3 stpr3 c2tpr3 s2tpr3  
  φ+ θ/2 ptd3 ptr3 cptr3 sptr3 c2ptr3 s2ptr3  
  2(θ+ φ/2) tpd3a tpr3a ctpr3a stpr3a c2tpr3a s2tpr3a  
  2(φ+ θ/2) ptd3a ptr3a cptr3a sptr3a c2ptr3a s2ptr3a  
  θ- φ/2 tpd4 tpr4 ctpr4 stpr4 c2tpr4 s2tpr4  
  φ- θ/2 ptd4 ptr4 cptr4 sptr4 c2ptr4 s2ptr4  
  2(θ- φ/2) tpd4a tpr4a ctpr4a stpr4a c2tpr4a s2tpr4a  
  2(φ- θ/2) ptd4a ptr4a cptr4a sptr4a c2ptr4a s2ptr4a  
  slope of st. line m1                      
  intercept of st.line on y-axis c1                      
  Integers       rotation θ (( r1) rotation φ (( R1) r1R1   R1r1   rotation θ+φ(rpR)  
  b b   (r11) (r12) (R11) (R12) (rR11) (rR12) (Rr11) (Rr12) (rpR11) (rpR12)
  a(a>b) a   (r21) (r22) (R21) (R22) (rR21) (rR22) (Rr21) (Rr22) (rpR21) (rpR22)
  a2+b2 t1   det: tr: det: tr: det: tr: det: tr: det: tr:
  √(a2+b2) t1a   λr1:  ±i* λR1:  ±i* λr1R1:  ±i* λR1r1:  ±i* λrpR:  ±i*
  -√(a2+b2) t1b   λ1+λ2: λ1λ2: λ1+λ2: λ1λ2: λ1+λ2: λ1λ2: λ1+λ2: λ1λ2: λ1+λ2: λ1λ2:
  2ab t2                      
  2a2b2 t2a   reflection θ (( re1) reflection φ (( Re1) re1Re1   Re1re1   rot2(θ-φ)(rmR)  
  a2-b2 t3   (re11) (re12) (Re11) (Re12) (reRe11) (reRe12) (Rere11) (Rere12) (rmR11) (rmR12)
  (a2+b2)2 t4   (re21) (re22) (Re21) (Re22) (reRe21) (reRe22) (Rere21) (Rere22) (rmR21) (rmR22)
  (a2-b2)2 t4a   det: tr: det: tr: det: tr: det: tr: det: tr:
  4ab(a2-b2) t5   λre1:  ±i* λRe1:  ±i* λr1R1:  ±i* λR1r1:  ±i* λrpR:  ±i*
  a2-b2+2ab t6a   λ1+λ2: λ1λ2: λ1+λ2: λ1λ2: λ1+λ2: λ1λ2: λ1+λ2: λ1λ2: λ1+λ2: λ1λ2:
  a2-b2-2ab t6b   rot2(φ-θ)(Rmr)   rotθ*refφ r1Re1 ref (φ -θ/2) Re1mre1 refφ*rotθ Re1r1 ref (φ +θ/2) Re1pre1
  t6a*t6b t6   (Rmr11) (Rmr12) (rRe11) (rRe12) (Remre11) (Remre12) (Rer11) (Rer12) (Repre11) (Repre12)
  t3/t1 t3c   (Rmr21) (Rmr22) (rRe21) (rRe22) (Remre21) (Remre22) (Rer21) (Rer22) (Repre21) (Repre22)
  t2/t1 t2c   det: tr: det: tr: det: tr: det: tr: det: tr:
  t6/t4 t6c   λrpR:  ±i* λ:  ±i* λR1r1:  ±i* λ:  ±i* λR1r1:  ±i*
  t5/t4 t5c   λ1+λ2: λ1λ2: λ1+λ2: λ1λ2: λ1+λ2: λ1λ2: λ1+λ2: λ1λ2: λ1+λ2: λ1λ2:
              rotφ*refθ R1re1 ref (θ -φ/2) re1mRe1        
              (Rre11) (Rre12) (remRe11) (remRe12)        
              (Rre21) (Rre22) (remRe21) (remRe22)        
              det: tr: det: tr:        
              λ:  ±i* λR1r1:  ±i*        
              λ1+λ2: λ1λ2: λ1+λ2: λ1λ2:        
          reflection around line y-mx+c                  
          (reli11) (reli12) (reli13)              
          (reli21) (reli22) (reli23)              
          (reli31) (reli32) (reli33)              
          det: tr:                
                             
                             
          integer rotation matrix   integer reflection matrix              
          (t3c) (t2c) (t6c) (t5c)            
          (-t2c) (t3c) (t5c) (-t6c)            
          det: tr: det: tr:            
          λ:  ±i* λ:  ±i*            
          λ1+λ2: λ1λ2: λ1+λ2: λ1λ2:            
                             
                             

 

 

* In the case of rotational matrix, the co-ordinate system remains fixed while the vector or tensor underlying an object is undergoing rotation. In the case of transformation matrix, the vector/tensor  remains fixed while the co-ordinate system undergoes rotation. Both the matrices are transpose of each other. Does transpose nature of matrix take care of chirality ?

 

*For rotation of a vector whose position vectors are (x,y) about the origin in anti-clockwise direction by an angle  θ while the reference frame remains fixed , the new position vectors are given by (x',y') where

x'=xcosθ - ysin θ ; y'=xsin θ + ycos θ & rotation vector is given by v'=Rv

R=[ cos  θ        -sin  θ

       sin  θ          cos  θ ]  

Calculation : x=l*cos φ, y=l*sin φ. On rotation anti-clockwise by θ,           x'=l*cos( φ + θ ) and y'= l*sin( φ + θ ) where l is the vector norm.

x'=l*cosφ cosθ -l*sinφ sinθ =xcosθ -ysinθ .Similarly , it can be proved that                                    y'=xsinθ +ycosθ

In our calculator as above, we have taken the rotation as clockwise with respect to the co-ordinate system.

* Let a reflection about a line through the origin which makes an angle  θ with the x-axis be ref( θ ). Then the corresponding reflection matrix Re( θ ) is given by

Re( θ ) = [ cos 2θ         sin 2θ

                 sin 2θ         -cos 2θ ]

rot(θ )*rot(φ) =rot(θ +φ)

re(θ )*re(φ)  =rot[2(θ-φ )]

rot(θ )*re(φ)=re(φ - θ/2 )

re(φ)*rot(θ )=re(φ + θ/2 ) (here, θ  & φ are clockwise)

* reflections about a line y=mx+c is given by the matrix

Re=[ cos2θ      sin2θ     -csin2θ

         sin2θ      -cos2θ   c(1+cos2θ)

          0              0                1        ] where θ is taken clockwise.

* The inverse of R(θ) =R(-θ). *The inverse of Re(θ)=Re(θ) . Reflection matrices are involutary.
* All rotation matrices are orthogonal matrices. The set of all 2x2 orthogonal matrices form a group O(2) under matrix multiplication. *Orthogonal matrices.
Determinant is +1. Hence only rotational matrices form special orthogonal group SO(2). However , rotational along with reflex matrices form O(2) since determinant can be +1 or -1, and they together form O(2). Determinant -1. form sub-group if identity matrix is included.
* Rotational Matrices have N(N-1)/2 independent parameters where N is the order/dimension of matrix. For N=2, the no. of independent parameters is 1 which is angle of rotation. For N=3, it is 3 which are the 3 angles of rotation around the basis vectors. In both the cases, the no. of independent parameters does not exceed the no. of dimensions. For N>3, the no. of independent parameters is more than the no. of dimensions. For N=4, no. of independent parameters equals 6.

We can look upon the whole thing like this. For rotation, one requires at the minimum, a plane which is 2-D. For any rotation in any dimension, no. of planes required is N C2 , N being the total no. of dimensions.

Rotational matrices correspond to active transformation.

 
*eigen values of rotational matrices are  cosθ  ± i*sin θ i.e. complex numbers except when angle is zero or 180 degree which implies all rotations in 2x2 real Hilbert Space yield complex eigen values. However , eigen value of Hermitian Operators(matrices) yield real eigen values

Suppose a matrix is given by [a  b

                                               -b  a]. The characterstic equation is (a-λ )2+b2=0. Solution is λ =a ib. If a=cosθ, b=sinθ, then the matrix becomes a rotation matrix.

* to find eigen vectors, R - λI = 0 or  [a  b  -  [ λ   0      =[ 0

                                                      -b  a]             0   λ ]       0] 

or [a-λ   b      *  [x    = [0

       -b   a-λ ]      y]        0]   

or  a1x+by =0  and bx+(a1-2a)y=0  where a1=a-λ

so y=-(a1/b)x=-[(a-λ)/b]x. Since λ=a ib, y=ix  or x=iy

*Suppose a matrix is given by [a  b

                                                  c  d]. The characterstic equation is (d-λ)(a-λ )-bc=0 or  λ2-(a+d)λ+(ad-bc)=0 or λ2-traceλ+determinant=0

or λ = tr/2   √[(tr/2)2 - det] =λa + λb where λa=tr/2 =av.trace and λb=√[(tr/2)2 - det] =√[(av.trace)2 - det]  and

 λ1 +λ2=trace(tr);

 λ1λ2 =determinant(det).

If eigen vector components are x,y, then y/x =(λ - a )/ b = c/(λ - d)

=1/b [ (d-a) /2  √[(tr/2)2 - det]] =component a component b where component a= (d-a)/2b=(av.anti-trace)/b ; component b= √[(tr/2)2 - det]] / b

or y =x[( λa-a)/b] x( λb/2b) = x*real part  x* real/imaginary part

We treat λ =λa + λb where λa=tr/2=av.trace and λb=√[(tr/2)2 - det]. suppose we want to make λb= √n2 (tr/2) = n (tr/2). The condition is

det =(tr/2)2 [1 - n2 ] ; for n=1, det=0        &       λb=0;

                                   for n=2, det=-3(tr/2)2   ;  λb=  2(tr/2);

                                   for n=3, det=-8 (tr/2)2   ; λb= 3(tr/2)

λb=√[(tr/2)2 - det]     λb =0 when   (tr/2)2 - det   =0 or

[6ad -(a2+d2)]/4 =bc        

λb > 0 when   (tr/2)2 - det  > 0 or

[6ad -(a2+d2)]/4 >bc    

λb < 0 when   (tr/2)2 - det  < 0 or

[6ad -(a2+d2)]/4 < bc 

In the case of eigen values,    λa is the fixed component and  λb is the variable component.  λa can be treated as a phase part.                

* Eigen values 1 i.e. real and constant irrespective of angle of rotation.

Suppose a matrix is given by [a  b

                                               b  -a]. The characterstic equation is λ2=(a2+b2) Solution is λ = √(a2+b2). If a=cosθ, b=sinθ, then the matrix becomes a reflection matrix.

* to find eigen vectors,

R - λI = 0 or  [a  b  -  [ λ   0      =[ 0

                      b  -a]     0   λ ]          0] 

or [a-λ   b      *  [x    = [0

      b   -a-λ ]      y]        0]   

or  a1x+by =0  and  bx+(a1-2a)y=0  where a1=a-λ

so y=-(a1/b)x=-[(a-λ)/b]x. Since λ= √(a2+b2), y= -[a √(a2+b2)]/b  *  x

For reflection matrix, a=cosθ, b=sinθ, y=(-a1)/b  * x  = tan(θ/2) *x  or y=-cot(θ/2) *x

* trace is 2cos θ whose values are [-2,2] * trace is zero
* sum of the eigen values=trace=2a=2cosθ. Product of the eigen values =cos2θ +sin2θ =1(determinant) * sum of the eigen values=0 and product of eigen values=determinant= -1
*Value of at least one rotation matrix for Rotation in n-dimensional Hilbert space is equal to reflection in n-1 dimensional space (hyperplane)  
* rotational matrices commute in even no. of dimensions under matrix multiplication and not in odd no. of dimensions * reflection matrices do not commute in general under matrix multiplication.
* form continuous symmetry and belong to Lie Group * form discrete symmetry and belong to discrete finite group.
*invariance under rotational symmetry is manifested as conservation of angular momentum *Invariance under reflection symmetry is manifested as conservation of parity.
*Rotational matrices are a sum of symmetric and skew symmetric matrices * Reflecting matrices are symmetric about the leading diagonal. By leading diagonal, we mean the elements from uppermost left hand corner to lowermost right hand corner.
*Two reflections in different planes is equivalent to a rotation.  
*rotational matrices produce similar triangles.  
*Rotational matrices are represented by [cosθ  -sinθ = [sin( π/2 - θ )  -sinθ

                                                                   sinθ    cosθ ]   sin θ        sin( π/2 -θ)]

can the rotations be construed  purely as change of phase (temporal part) from θ =0 to θ=θ without any spatial movement.

 

*Eigen vectors of all 2-D rotational matrices are[1,i], [1,-i] or [i,1],[-i,1] hence, they(rotation matrices) commute. If we apply the formula,

y/x =(λ - a )/ b =1/b [ (d-a) /2  √[(tr/2)2 - det]] here where a=c=cosθ, b=sinθ=-d, we get y/x=i or y=ix ; (The contribution of component a to the eigen vector ratio is zero.)

                                                           If x=1, y= i  and y=-i

                                                           if y=1, x=i   and x=-i

one interesting thing as observed above is that (x,y) are perfectly interchangeable and nature does not distinguish between both the axis.The choice is arbitrary.  If you replace x by y or vice versa, the relation remains invariant. Moreover x2 =-y2  and y2 = -x2 . or x2 + y2 =0 ; The equation indicates that in real Hilbert space, if one co-ordinate is spatial co-ordinate and the other is a time co-ordinate , the Minokowski 2-D space-time equation follows from rotational matrix eigen vectors, provided we take c=1, t=y i.e. if we put y=ict, then  x2 - c2t2 =0

 
If a, b are 2 integers, a >b, then the rotation matrix is given by

(1/ (a2+b2)  )[a2 - b2      2ab

                      -2ab     a2 - b2  ] where t1=(a2+b2), t2=2ab, t3= a2 - b2

 = 1/t1[t3  t2

          -t2,  t3] = [t3c   t2c

                          -t2c  t3c ] t1,t2,t3 are Pythagorean triplets

eigen value = λ = t3c i*t2c

If a, b are 2 integers, a >b, then the reflection  matrix is given by

(1/t12) [t32 - t22      2t2t3

             2t2t3     -(t32 - t22)] = (1/t4) [ t6   t5     = [ t6c   t5c

                                                             t5   -t6]        t5c  -t6c ]

eigen value = √(t6c2 +t5c2) = 1

* The dot product of any row or column with itself equals one.                      (cos θ*i+sinθ *j)*(cosθ *i+sin θ*j)=1

* The dot product of any row or column with any other row or column is zero. (cos θ*i+sinθ *j)*(cosθ *j-sin θ*i)=0

* In 3-D, the rotation matrix is

R[cos(x',x)     cos(y',x)  cos(z',x)

    cos(x',y)     cos(y',y)  cos(z',y)

    cos(x',z)     cos(y',z)  cos(z',z) ]

If the rotation is about say z-axis, then cos(z',z)=1 and cos(x',z)=cos(y',z) =cos(z',x)=cos(z',y)=cos 90 degree =0. cos(y',x)=cos(90+θ)=-sinθ ; cos(x',y) = cos(90-θ)=sinθ. And matrix becomes

[cosθ  -sinθ   0

 sinθ   cosθ    0

  0        0        1] and 2-D matrix is revealed within 3-D matrix.

 
   
   
 

Parity Operators & Matrices: Parity involves an operation that changes the algebric sign of the co-ordinates. Thus a point (x,y,z) shall be inverted to (-x,-y,-z) under parity transformation. Parity operations are discrete operations and hence have no generators. If P is the parity matrix, P2 =I where I is the unit matrix. If Ψ(x) is a wave function, then P|Ψ(x) > = -|Ψ(x) > . P commutes with the Hamiltonian H of the system. P is an involutary matrix. So its eigen value is +1(even) or -1(odd). P is also hermitian. In 2x2 matrix, P= [-1 0

                                                                                                                                 0 -1]

In 1-D, parity is same as reflection in a point size mirror placed at the origin.

In 2-D, parity is same as 180 degree rotation about the origin or reflection about a line passing through origin and perpendicular to the line connecting the point and the origin.

In 3-D, parity is more than just a reflection in a plane mirror. It is equivalent to reflection followed by rotation . For example, mirror is in x-y plane, then z-axis -> -z which implies right handed co-ordinate system is transferred to left handed co-ordinate system.

 Normally, wave functions that describe particles should not behave in a different way if we choose a right handed or left handed co-ordinate system. But they do so in weak interactions such as beta decay. Let us choose the decay of Co60  to Ni60 and electron, anti-neutrino.

 Co60 ->  Ni60 +e- + anti-νe

Madam Wu , who conducted the experiment lowered the temperature of cobalt atoms and was able to polarize the nuclear spins along the direction of applied magnetic field. The directions of emitted electrons were then measured. Equal no. of electrons to be emitted parallel and anti parallel to the external magnetic field if parity was conserved. But, it was found that more no. of electrons were emitted anti-parallel to the magnetic field and therefore opposite to the nuclear spin. This resulted in the discovery of neutrinos. Anti-nuetrinos have their spin parallel to their velocity and are called right handed particles. The very idea that the nature can distinguish a right handed system from a left handed one is a very radical idea.

 

Look at youself in a mirror and you will see the parity reflection of what other people see. Space inversion, basically.Parity was taken to be an absolute symmetry in physics until Lee and Yang questioned it to solve a problem which seemed to contradict the normal wisdom. They went as far as to suggest an experiment to another chinese researcher, Mrs Wu who performed the experiment and confirmed their suggestion: the so-called weak interactions do indeed violate parity. Later it was found that they violate also Charge conjugation and also Time invariance.

A friend of mine working in the nearby office (he later collaborated with Lee and Yang on a related subject) was telling me that they were working and talking in chinese except that very often the word "parity" would pop in during their animated discussions.

One formal way to think of Parity is as a transformation. Parity acts on objects and, in general, changes them. Its basic action is quite easy to visualize: Take point r in space to point -r. Then, after a parity transformation with origin the center of earth, all points in the "sphere" of Earth will be where their antipodes are right now. Objects that do not change, like a billiard ball, are special with respect to the parity transformation.
Some objects change so much, that they become very different objects with very different properties. Like for example a twisted cork opener. After a parity transformation, the sense of screw has changed and you would have to use it differently for opening a bottle. Other objects change, but not in essence, like a cone, which appears as a copy of itself after parity. Nothing as dramatic as what happened to the cork opener.
Similar ideas apply when parity acts on other objects, like electrons, photons, etc ... . Some do not change at all, some change but not fundamentally, and some change in a fundamental way, like for example the fact that neutrinos come in only one handedness. So, left and right handed neutrinos are fundamentally different (one of them exists and the other does not) , and transform into each other through the parity transformation. Other objects, like the sum of two photons of different handedness (contain both screw senses in them) do not fundamentally change after parity (each of the two screw senses turns into the other one and thing stay the same).
The way stuff transforms after a parity operation is used as a means for classifying physical objects in a systematic way and, often, to learn things about how they interact with other objects