Rotation & Reflection Matrix
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* 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 ? |
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*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. |
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| *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)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=(-a±1)/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. |
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*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 |
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| 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. |
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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
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