Examples of 2x2 matrices & Orthogonality
1a 1b |
x (a) y(b)
+ + y(c) x(d) + + |
x (a) -y(b) +
- -y(c) x(d) - + |
1c 1d |
x (a) -y(b) +
- y(c) x(d) + + |
x (a) y(b) + + -y(c) x(d) - + |
|
Δ =x2 -y2 | Δ =x2 -y2 | Δ =x2 + y2 | Δ =x2 + y2 | |||
tr =2x | tr =2x | tr =2x | tr =2x | |||
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
|||
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
|||
condition 2a: ab=-cd not satisfied ab=cd satisfied. |
condition 2a: ab=-cd not satisfied ab=cd satisfied. |
condition 2a: ab=-cd
satisfied |
condition 2a: ab=-cd satisfied |
|||
condition 2b: ac=-bd not satisfied ac=bd satisfied. |
condition 2b: ac=-bd not satisfied ac=bd satisfied. |
condition 2b: ac=-bd satisfied |
condition 2b: ac=-bd satisfied |
|||
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy. |
condition 3:a2 +b2 =1=c2 +d2 *can be made to satisfy. In that case, a2 +b2 =1 is the equation of a unit radius circle as well as pythogorean equation of a right angled triangle with hypotenuse as unity. |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
|||
not orthogonal; akin to rotational matrix- AT =A matrix(1a)' =matrix (1b) [1a,1b]=0 |
not orthogonal; akin to rotational matrix- AT =A matrix(1b)' =matrix (1a) [1b,1a]=0d |
orthogonal(rotation) matrix(1c)-1 =matrix(1d) |
orthogonal(rotation) matrix(1d)-1 =matrix(1c) |
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eigen value: x ±y e.vector (y/x): ±1 ev1=i' +j' ev2=i' - j' <ev1,ev2> =1-1=0
|
eigen value: x ±y e.vector (y/x):±1 ev1=i' +j' ev2=i' - j' <ev1,ev2> =1-1=0
|
eigen value: x ±iy e.vector (y/x):(y/y)=±i ev1=i' +i*j' ev2=i' -i*j' <ev1,ev2> =1-1=0 |
eigen value: x ±iy e.vector (y/x):(y/y)=±i ev1=i' +i*j' ev2=i' -i*j' <ev1,ev2> =1-1=0 |
|||
2a 2b |
x (a) y(b)
+ + -y(c) -x(d) - - |
x (a) -y(b) + - y(c) -x(d) + - |
2c 2d |
x (a) y(b) + + y(c) -x(d) + - |
x (a) -y(b) +
- -y(c) -x(d) - - |
|
Δ =-(x2 - y2) | Δ =-(x2 - y2) | Δ =-x2 - y2 | Δ =-x2 - y2 | |||
tr =zero | tr =zero | tr =zero | tr =zero | |||
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
|||
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
|||
condition 2a: ab=-cd not satisfied |
condition 2a: ab=-cd not satisfied |
condition 2a: ab=-cd satisfied |
condition 2a: ab=-cd satisfied |
|||
condition 2b: ac=-bd not satisfied |
condition 2b: ac=-bd not satisfied |
condition 2b: ac=-bd satisfied |
condition 2b: ac=-bd satisfied |
|||
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
|||
not orthogonal; akin to reflection matrix- A-1 =A, involutary matrix if Δ =-1.But this can happen only when the off -diagonal elements are imaginary. matrix(2a)' =matrix (3b) [2a,3b]=0
|
not orthogonal; akin to reflection matrix- A-1 =A, involutary matrix if Δ =-1. But this can happen only when the off -diagonal elements are imaginary. matrix(2b)' =matrix (3a) [2b,3a]=0
|
orthogonal(reflection) involutary as AT =A |
orthogonal(reflection) involutary as AT =A |
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eigen value: ±√(x2 -y2) e.vector (y/x): (-x/y)±√(x2 -y2)/y = (-x/y)±√(x2/4 - 1)
|
eigen value: ±√(x2 -y2) e.vector (y/x): (-x/y)±√(x2 -y2) / y |
eigen value: ±√(x2 +y2) e.vector (y/x): (-x/y)±√(x2 +y2) / y taking x=cosθ ; y=sinθ (y/x) =-cotθ ±cosecθ ev1=i' +(-cotθ+cosecθ)j'= i'+tan(θ/2)j' ev2=i' +(-cotθ-cosecθ)j'= i'-cot(θ/2)j' <ev1,ev2>=1-1=0 |
eigen value: ±√(x2 +y2) e.vector (y/x): (x/y)±√(x2 +y2) / y taking x=cosθ ; y=sinθ (y/x) =cotθ ±cosecθ ev1=i' +(cotθ+cosecθ)j'= i'+cot(θ/2)j' ev2=i' +(cotθ-cosecθ)j'= i'- tan(θ/2)j' <ev1,ev2>=1-1=0 |
|||
3a 3b |
-x (a) y(b)
- + -y(c) x(d) - + |
-x (a) -y(b)
- - y(c) x(d) + + |
3c 3d |
-x (a) y(b) -
+ y(c) x(d) + + |
-x (a) -y(b)
- - -y(c) x(d) - + |
|
Δ =-(x2 - y2) | Δ =-(x2 - y2) | Δ =-x2 - y2 | Δ =-x2 - y2 | |||
tr =zero | tr =zero | tr =zero | tr =zero | |||
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
|||
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
|||
condition 2a: ab=-cd not satisfied |
condition 2a: ab=-cd not satisfied |
condition 2a: ab=-cd satisfied |
condition 2a: ab=-cd satisfied |
|||
condition 2b: ac=-bd not satisfied |
condition 2b: ac=-bd not satisfied |
condition 2b: ac=-bd satisfied |
condition 2b: ac=-bd satisfied |
|||
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
|||
not orthogonal; akin to reflection matrix- A-1 =A, involutary matrix if Δ =-1.But this can happen only when the off -diagonal elements are imaginary. matrix(3a)' =matrix (2b) [3a,2b]=0
|
not orthogonal; akin to reflection matrix- A-1 =A, involutary matrix if Δ =-1. But this can happen only when the off -diagonal elements are imaginary. matrix(3b)' =matrix (2a) [3b,2a]=0
|
orthogonal(reflection) Involutary as AT =A |
orthogonal(reflection) Involutary as AT =A |
|||
eigen value: ±√(x2 -y2) e.vector (y/x): (x/y)±√(x2 -y2) / y |
eigen value: ±√(x2 -y2) e.vector (y/x): (-x/y)±√(x2 -y2) / y |
eigen value: ±√(x2+y2) e.vector (y/x): (x/y)±√(x2 +y2) / y taking x=cosθ ; y=sinθ (y/x) =cotθ ±cosecθ ev1=i' +(cotθ+cosecθ)j'= i'+cot(θ/2)j' ev2=i' +(cotθ-cosecθ)j'= i'- tan(θ/2)j' <ev1,ev2>=1-1=0
|
eigen value: ±√(x2+y2) e.vector (y/x): (-x/y)±√(x2 +y2) / y taking x=cosθ ; y=sinθ (y/x) =-cotθ ±cosecθ ev1=i' +(-cotθ+cosecθ)j'= i'+tan(θ/2)j' ev2=i' +(-cotθ-cosecθ)j'= i'-cot(θ/2)j' <ev1,ev2>=1-1=0 |
|||
4a 4b |
-x (a) y(b) -
+ y(c) -x(d) + - |
-x (a) -y(b)
- - -y(c) -x(d) - - |
4c 4d |
-x (a) y(b)
- + -y(c) -x(d) - - |
-x (a) -y(b) - - y(c) -x(d) + - |
|
Δ =x2 -y2 | Δ =x2 -y2 | Δ =x2 + y2 | Δ =x2 + y2 | |||
tr =-2x | tr =-2x | tr =-2x | tr =-2x | |||
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
|||
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
|||
condition 2a: ab=-cd not satisfied ab=cd satisfied. |
condition 2a: ab=-cd not satisfied ab=cd satisfied. |
condition 2a: ab=-cd satisfied |
condition 2a: ab=-cd satisfied |
|||
condition 2b: ac=-bd not satisfied ac=bd satisfied. |
condition 2b: ac=-bd not satisfied ac=bd satisfied. |
condition 2b: ac=-bd satisfied |
condition 2b: ac=-bd satisfied |
|||
condition 3:a2 +b2 =1 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
|||
not orthogonal; akin to rotational matrix- AT =A matrix(4a)' =matrix (4b) [4a,4b]=0 |
not orthogonal ; akin to rotational matrix- AT =A matrix(4b)' =matrix (4a) [4b,4a]=0
|
orthogonal(rotation) | orthogonal(rotation) | |||
eigen value:-x ±y e.vector (y/x): ±1 ev1=i' +j' ev2=i' - j' <ev1,ev2> =1-1=0 |
eigen value:-x ±y e.vector (y/x): ±1 ev1=i' +j' ev2=i' - j' <ev1,ev2> =1-1=0
|
eigen value:-x ±iy e.vector (y/x): ±i ev1=i' +i*j' ev2=i' -i*j' <ev1,ev2> =1-1=0 |
eigen value:-x ±iy e.vector (y/x): ±i ev1=i' +i*j' ev2=i' -i*j' <ev1,ev2> =1-1=0 |
|||
5a 5b |
y (a) x(b)
+ + x(c) y(d) + + |
y (a) -x(b) +
- -x(c) y(d) - + |
5c 5d |
y (a) -x(b) +
- x(c) y(d) + + |
y (a) x(b) + + -x(c) y(d) - + |
|
Δ =y2 -x2 | Δ =y2 -x2 | Δ =y2 + x2 | Δ =y2 + x2 | |||
tr =2y | tr =2y | tr =2y | tr =2y | |||
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
|||
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
|||
condition 2a: ab=-cd not satisfied ab=cd satisfied. |
condition 2a: ab=-cd not satisfied ab=cd satisfied. |
condition 2a: ab=-cd satisfied |
condition 2a: ab=-cd satisfied |
|||
condition 2b: ac=-bd not satisfied ac=bd satisfied. |
condition 2b: ac=-bd not satisfied ac=bd satisfied. |
condition 2b: ac=-bd satisfied |
condition 2b: ac=-bd satisfied |
|||
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
|||
not orthogonal; akin to rotational matrix- AT =A matrix(5a)' =matrix (5b) [5a,5b]=0
|
not orthogonal; akin to rotational matrix- AT =A matrix(5b)' =matrix (5a) [5b,5a]=0
|
orthogonal(rotation) | orthogonal(rotation) | |||
eigen value:y ±x e.vector (y/x): ±1 ev1=i' +j' ev2=i' - j' <ev1,ev2> =1-1=0
|
eigen value:y ±x e.vector (y/x): ±1 ev1=i' +j' ev2=i' - j' <ev1,ev2> =1-1=0 |
eigen value:y ±ix e.vector (y/x): ±i ev1=i' +i*j' ev2=i' -i*j' <ev1,ev2> =1-1=0 |
eigen value:y ±ix e.vector (y/x): ±i ev1=i' +i*j' ev2=i' -i*j' <ev1,ev2> =1-1=0 |
|||
6a 6b |
y (a) x(b)
+ + -x(c) -y(d) - - |
y (a) -x(b) + - x(c) -y(d) + - |
6c 6d |
y (a) x(b) + + x(c) -y(d) + - |
y (a) -x(b) +
- -x(c) -y(d) - - |
|
Δ =-(y2 - x2) | Δ =-(y2 - x2) | Δ =-y2 - x2 | Δ =-y2 - x2 | |||
tr =zero | tr =zero | tr =zero | tr =zero | |||
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
|||
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
|||
condition 2a: ab=-cd not satisfied |
condition 2a: ab=-cd not satisfied |
condition 2a: ab=-cd satisfied |
condition 2a: ab=-cd satisfied |
|||
condition 2b: ac=-bd not satisfied |
condition 2b: ac=-bd not satisfied |
condition 2b: ac=-bd satisfied |
condition 2b: ac=-bd satisfied |
|||
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
|||
not orthogonal; akin to reflection matrix- A-1 =A, involutary matrix if Δ =-1.But this can happen only when the off -diagonal elements are imaginary. matrix(6a)' =matrix (7b) [6a,7b]=0
|
not orthogonal; akin to reflection matrix- A-1 =A, involutary matrix if Δ =-1.But this can happen only when the off -diagonal elements are imaginary. matrix(6b)' =matrix (7a) [6b,7a]=0
|
orthogonal(reflection) Involutary as AT =A |
orthogonal(reflection) Involutary as AT =A |
|||
eigen value:
±√(y2 -x2) e.vector (y/x): (-y/x)±√(y2 -x2) / x |
eigen value: ±√(y2
-x2) e.vector (y/x): (y/x)±√(y2 -x2) / x
|
eigen value: ±√(y2+x2) e.vector (y/x): (-y/x)±√(y2+x2) / x taking x=cosθ ; y=sinθ (y/x) =-tanθ ±secθ ev1=i' +(-tanθ+secθ)j' ev2=i' +(-tanθ-secθ)j' <ev1,ev2>=1 - (sec2θ - tan2θ )=1 -1=0 |
eigen value: ±√(y2+x2) e.vector (y/x): (y/x)±√(y2+x2) / x taking x=cosθ ; y=sinθ (y/x) =tanθ ±secθ ev1=i' +(tanθ+secθ)j' ev2=i' +(tanθ-secθ)j'=i' -(secθ - tanθ) <ev1,ev2>=1 - (sec2θ - tan2θ )=1 -1=0 |
|||
7a 7b |
-y (a) x(b)
- + -x(c) y(d) - + |
-y (a) -x(b)
- - x(c) y(d) + + |
7c 7d |
-y (a) x(b) -
+ x(c) y(d) + + |
-y (a) -x(b)
- - -x(c) y(d) - + |
|
Δ =-(y2 - x2) | Δ =-(y2 - x2) | Δ =-y2 - x2 | Δ =-y2 - x2 | |||
tr =zero | tr =zero | tr =zero | tr =zero | |||
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
|||
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
|||
condition 2a: ab=-cd not satisfied |
condition 2a: ab=-cd not satisfied |
condition 2a: ab=-cd satisfied |
condition 2a: ab=-cd satisfied |
|||
condition 2b: ac=-bd not satisfied |
condition 2b: ac=-bd not satisfied |
condition 2b: ac=-bd satisfied |
condition 2b: ac=-bd satisfied |
|||
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
|||
not orthogonal; akin to reflection matrix- A-1 =A, involutary matrix if Δ =-1.But this can happen only when the off -diagonal elements are imaginary. matrix(7a)' =matrix (6b) [7a,6b]=0
|
not orthogonal; akin to reflection matrix- A-1 =A, involutary matrix if Δ =-1.But this can happen only when the off -diagonal elements are imaginary. matrix(7b)' =matrix (6a) [7b,6a]=0
|
orthogonal(reflection) Involutary as AT =A |
orthogonal(reflection) Involutary as AT=A |
|||
eigen value:
±√(y2 -x2) e.vector (y/x): (y/x)±√(y2 -x2) / x |
eigen value:
±√(y2 -x2) e.vector (y/x): (-y/x)±√(y2 -x2) / x |
eigen value:
±√(y2 +x2) e.vector (y/x): (y/x)±√(y2 +x2) / x taking x=cosθ ; y=sinθ (y/x) =tanθ ±secθ ev1=i' +(tanθ+secθ)j'=i' + (sin θ/2+cos θ/2) / (-sin θ/2+cos θ/2) =i' + (1+tan θ/2) / (1-tan θ/2) ev2=i' +(tanθ-secθ)j'=i' -(secθ - tanθ)=i' - (-sin θ/2+cos θ/2) / (sin θ/2+cos θ/2) =i' - (1-tan θ/2) / (1+tan θ/2) <ev1,ev2>=1 - (sec2θ - tan2θ )=1 -1=0 |
eigen value:
±√(y2 +x2) e.vector (y/x): (-y/x)±√(y2 +x2) / x taking x=cosθ ; y=sinθ (y/x) =-tanθ ±secθ ev1=i' +(-tanθ+secθ)j' ev2=i' +(-tanθ-secθ)j' <ev1,ev2>=1 - (sec2θ - tan2θ )=1 -1=0 |
|||
8a 8b |
-y (a) x(b) -
+ x(c) -y(d) + - |
-y (a) -x(b)
- - -x(c) -y(d) - - |
8c 8d |
-y (a) x(b)
- + -x(c) -y(d) - - |
-y (a) -x(b) - - x(c) -y(d) + - |
|
Δ =y2 -x2 | Δ =y2 -x2 | Δ =y2 + x2 | Δ =y2 + x2 | |||
tr =-2y | tr =-2y | tr =-2y | tr =-2y | |||
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
condition 1a: a2 +b2 =c2 +d2
satisfied |
|||
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
condition 1b: a2 +c2 =b2 +d2
satisfied |
|||
condition 2a: ab=-cd not satisfied ab=cd satisfied. |
condition 2a: ab=-cd not satisfied ab=cd satisfied. |
condition 2a: ab=-cd satisfied |
condition 2a: ab=-cd satisfied |
|||
condition 2b: ac=-bd not satisfied ac=bd satisfied. |
condition 2b: ac=-bd not satisfied ac=bd satisfied. |
condition 2b: ac=-bd satisfied |
condition 2b: ac=-bd satisfied |
|||
condition 3:a2 +b2 =1 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
condition 3:a2 +b2 =1=c2
+d2 *can be made to satisfy |
|||
not orthogonal; akin to rotational matrix- AT =A matrix(8a)' =matrix (8b) [8a,8b]=0 |
not orthogonal ; akin to rotational matrix- AT =A matrix(8b)' =matrix (8a) [8b,8a]=0 |
orthogonal(rotation) | orthogonal(rotation) | |||
eigen value:-y±x Eigen vector(y/x):±1 ev1=i' +j'
ev2=i' - j' <ev1,ev2> =1-1=0 |
eigen value:-y±x Eigen vector(y/x):±1 ev1=i' +j' ev2=i' - j' <ev1,ev2> =1-1=0 |
eigen value:-y±ix Eigen vector(y/x):±i ev1=i' +i*j' ev2=i' -i*j' <ev1,ev2> =1-1=0 |
eigen value:-y±ix Eigen vector(y/x):±i ev1=i' +i*j' ev2=i' -i*j' <ev1,ev2> =1-1=0 |
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structure of orthogonal matrix:2x2 | sign
no. of matrices orthogonal ++++(4) 1 No +++ (3) 4 Yes ++ (2) 6 No + (1) 4 Yes (0) 1 No Total 16 There are alternate No & Yes starting with 4 plus signs by flipping successive plus with minus.The figures are given below:- |
For orthogonality, there should be 3 + or 3 - i.e. 3 plus signs & 1 minus sign or 3 minus signs & 1 plus sign. | link:1, | |||
±x(a) ± √(1-x2) (b) | ∓x(a) ∓ √(1-x2) (b) | * no. of variables is 1. |
*if part of condition 3:1=a2 +b2 =c2 +d2 is not satisfied , i.e a2 +b2 =c2 +d2 but they are not equal to 1,then AAT = scalar matrix in stead of unit matrix.In this case, A matrix can be made orthogonal by dividing each matrix element by square root of the determinant which is nothing but normalization. |
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∓√(1-x2) (c) ± x (d) | ∓√(1-x2) (c) ± x (d) | * determinant Δ ± 1 | * condition 1,2 imply that if a=d, then b=-c and Δ =+1; if a=-d,then b=c & Δ=-1. | |||
In fact , the structure of an orthogonal matrix can be constructed where a and b can be interchanged along with interchange of c and d.
tr=±2x with range [-2,2] (rotation matrix) determinant:+1 eigen value: * x + i√(1-x2) * x - i√(1-x2) * -x + i√(1-x2) * -x - i√(1-x2)
If x=cosθ , eigen values are cosθ +i sinθ = eiθ cosθ -i sinθ = e-iθ -cosθ +i sinθ = -e-iθ -cosθ -i sinθ = -eiθ If x=sinθ , eigen values are sinθ +i cosθ = ie-iθ sinθ -i cosθ = -ieiθ -sinθ +i cosθ = ieiθ -sinθ -i cosθ = -ie-iθ *In rotational matrix, a2 + b2=1 which is the equation of a set of right angled triangles with hypotenuse 1 and also the equation of a circle with unit radius. * ratio of y-component to x-component of eigen vector y/x =±i The vectors are A B C D (1 or (i or (1 or (-i i ) 1) -i) 1)
E F G H (-1 or (i or (-i or (-1 i ) -1) -1) -i)
A.C =1-i2 =2 B.D =1+1 =2 A.B=i+i=2i C.D=-2i B.C=i-i=0 A.D=-i+i=0 Among A to D, no. of arrangements 4C2=6 |A|=B|=|C|=D| =√2 E.G=+i-i=0 E.F=-i-i=-2i F.G=-i2+1=2 F.H=-i+i=0 G.H=i+i=2i E.H=1+1=2 |E|=F|=|G|=H| =√2 Among E to H, no. of arrangements 4C2=6 A.E=-1+i2 =-2 A.F=i-i=0 A.G=-i-i=-2i A.H=-1+1=0 B.E=-i+i=0 B.F=i2-1=-2 B.G=-i2-1=0 B.H=-i-i=-2i C.E=-1-i2=0 C.F=i+i=2i C.G=-i+i=0 C.H=-1+i2 =-2 D.E=i+i=2i D.F=-i2+1=0 D.G=i2-1=-2 D.H=i-i=0 From (A to D) to (E to H) , no. of arrangement 4*4=16 total arrangement:6+6+16= 28. Now we define pure real pairs as ones whose numerator (X.Y) is a real number. Exa-AC,BD,FG,EH - numerator +2 AE,BF,DG,CH- numerator is2 Now we define pure imaginary pairs as ones whose numerator (X.Y) is an imaginary number. Exa- AB,GH,CF,DE numerator: +2i CD,EF,AG,BH numerator: -2i Now if X,Y are 2 vectors , angle θ between them is given by cos θ = X.Y /[|X|*|Y|] So for all above pair of vectors, denominator is 2 and numerator are the respective dot products. For A&C, B&D,A&B -angle is 0° ; For F&G, E&H,G&H -angle is 0° ; For C&D -angle is 180° ; For E&F -angle is 180° ; For B&C,A&D -angle is 90° ; For E&G,F&H -angle is 90° ; For C&F,D&E-angle is 0° ; For A&F,A&H,B&E,B&G,C&E, C&G,D&F,D&H -angle is 90° ; For A&E,B&F,D&G,C&H,A&G, B&H -angle is 180° ; * the 12 pairs of eigen vectors are orthogonal to each other as the product of their slope is -1 *8 pairs of eigen vectors are parallel. *8 pairs of eigen vectors are anti-parallel. Ratio - 8:12:8=2:3:2 * Out of parallel vectors, numerator (X.Y)of 4 are real no. * Out of anti-parallel vectors, numerator of 4 (X.Y)are real no. *numerator of 12(X.Y) orthogonal vectors zero Ratio-4:12:4=1:3:1 Transition of vector pairs from angle zero to angle 180 with real denominator of X.Y AC->CH ; passing through AH BD->DG;passing through BG GF->FB;passing through BG HE->EA;passing through AH Transition of vector pairs from angle zero to angle 180 with imaginary denominator of X.Y AB->BH ; passing through AH DE->EF;passing through DF HG->GA;passing through AH FC->CD;passing through FD These are pure transitions from real to real denominators from angle zero to 180 or from imaginary to imaginary from angle zero to 180. Passing through vector pairs are having angle 90. AH pair is very peculiar as it appears both real-real and imaginary -imaginary transitions. Out of the 8+12+8 pairs, 8+(3)+8 are mapped, the 3 being AH,BG,DF. Now out of 12 with zero denominator , 9 are left unmapped. Transition of vector pairs from angle 90 to angle 90 with zero denominator of X.Y AD->DH ..passing through AH AF->FH ..passing through AH AH is already involved in real-real, imaginary-imaginary and now zero-zero transitions. CB->BE..passing through CE CG->GE..passing through CE ----------------------------- * slope of each vector to the axis can be measured from y/x * similarity matrix for the above orthogonal matrix is- 1 1 i -i *If vector A=ai +cj & vector B=bi+dj , then if matrix R = a b c d is orthogonal, then A,B are linearly independent i.e. their inner product is zero. <A> . <B> =0 & <A> .<A> =1 <B> . <B>=1 |
tr=0 (reflection matrix) determinant:-1 eigen value: * +1 * -1 * ratio of y-component to x-component of eigen vector y/x =[ -x/√(1-x2)] ± [1/√(1-x2)] =√ [(1-x)/(1+x)] and y/x=- √ [(1+x)/(1-x)] * the 2 eigen vectors are orthogonal in case of rotational matrices: Proof: Let A= i' +ij' =1*i'+1*ij' then B= i'- ij' =1*i'+(-1)ij' where i',j' are unit vectors along x,y axis respectively, i is imaginary number. <A,B>=1*1+1*(-1)=0 hence cosθ =0 and θ=π /2
Hence they are perpendicular. * The 2 eigen vectors are perpendicular to each other in reflection matrix. Proof: Let A=i' + tanθj' then B=i' - cotθj' Angle between the vectors given by cosθ = A.B / |A||B|= 0/|A||B|=0 Hence they are perpendicular.
* slope of each vector to the axis can be measured from y/x * similarity matrix for the above matrix is -- 1 1 √ [(1-x)/(1+x)] - √ [(1+x)/(1-x)] *If orthogonal matrix is represented as [A]= cosθ -sinθ sinθ cosθ √[A]= cos(θ/2) -sin(θ/2) sin(θ/2) cos(θ/2) (to prove use cosθ =cos2θ/2-sin2θ/2 & sinθ =2sinθ/2cosθ/2 ) Similarly If orthogonal matrix is represented as [A]= a -√(1-a2) √(1-a2) a , then √[A]= √[(1+a)/2] -√[(1-a)/2] √[(1-a)/2] √[(1+a)/2] Circular Rotation: cosθ -sinθ * x = x' sinθ cosθ y y' x'2 + y'2 =x2 +y2 x'2-y'2=cos2θ(x2-y2)-sin2θ(2xy) x'2-y'2=0 when tan2θ=(x2-y2) /2xy Further, x'2-y'2=cos2θ(x2-y2)-sin2θ(2xy) = (x2+y2)[cos2θ*(x2-y2)/(x2+y2) - sin2θ*2xy/(x2+y2)] = (x2+y2)[cos2θ*sin2φ -sin2θ*cos2φ] x'2-y'2=(x2+y2)[sin2(φ-θ)] where sin2φ =(x2-y2)/(x2+y2) cos2φ =(2xy)/(x2+y2) Eigen Vector Component ratio y/x in terms of reflection angle (y/x) part-1 : anti-trace/2b =∓2cos2θ/±2sin2θ=-cot2θ (y/x) part-2 :±cosec2θ (y/x): -cot2θ +cosec2θ =±tanθ and (y/x): -cot2θ -cosec2θ =∓cotθ while determining the angle, one needs to draw diagram because arctan & arccot throw results in first quadrant only. --------------------------------- * product of 2 orthogonal matrices is orthogonal and product of 2 unitary matrices is unitary. * Matrix A is diagonalizable with a unitary matrix iff it is normal. * For orthogonal matrices, eigen vectors corresponding to different eigen values are orthogonal. All orthogonal matrices are normal matrices. And normal matrices can be diagonalized by unitary matrices. These unitary matrices can be constructed from ortho-normal eigen vectors. The converse is also true. * While considering rotational matrices, we assume that co-ordinate system is right handed i.e. positive x-axis is to the right of positive y-axis and when an observer views the rotation in anti-clockwise direction, it is positive. Rotation matrix is to the left of the vector when pre-multiplied. * If A is a rotational matrix, then (A-I)*(A+I)-1 is a skew symmetric matrix. Hence from any skew-symmetric matrix say B, we can construct a rotational matrix by premultiplying with (I+B)(I-B)-1 . Such skew-symmetric matrix shall have n(n-1)/2 independent numbers where n is the order of the matrix. * Many copies of n-dimensional rotations are found in (n+1) dimensional rotation as sub-groups. Such embedding leaves one direction fixed which is the rotation axis in 3x3 matrices. *suppose there is a 2x2 real matrix A= x -y y x where √(x2 +y2) =r and not 1, then we can express A as a product of rotation matrix & a scaling matrix- cosθ -sinθ * r 0 sinθ cosθ 0 r if |r| > 1, then repeated application of the operator on 2-d vectors results in spiraling out. if |r| < 1, then repeated application of the operator on 2-d vectors results in spiraling in. if |r| = 1, then repeated application of the operator on 2-d vectors results in vectors moving in an elliptical path. r is called the scaling factor. |
*In orthogonal matrices, AAT=I . If A= a b c d then AT = a c b d Since A ,AT are inter changeable, there is a fuzziness about (b,c). When there is rotational oscillation, the system appears achiral and any transformation involving orthogonal matrix has an inbuilt chirality with invariance of vector norm. Only for vectors with specific y/x ratio, rotational matrix produces achiral vector i.e no change in direction but norm is not preserved , it is either stretched or shrinks. But position vector of one point of the vector remains the same (the origin) where as the other point linearly shifts. In normal rotation, the same point angularly shifts. Try to find a situation where both norms remain the same and direction also is not changed. The most obvious ones are the reflection matrices which keep y/x ratio the same and modulus of eigen value is 1. *matrix elements, a,b,c,d are less than/equal to +1 and greater than/equal to -1, i.e range is [-1,1] and real. a ± b= a ±√ (1- a2 ) = √(1 ±2a√[1- a2] ) =√(1±sin2θ ) by taking a =cosθ and then b=sinθ and a ± b is in the range [0,√2] & [-√2 , 0] * In orthogonal matrices, trace curve is a sin/cos curve amplified by a factor 2. |
condition of Orthogonality for purely imaginary numbers * if all the elements of the orthogonal matrix (a,b,c,d) are imaginary numbers, it follows that a2 +b2 =-1=c2 +d2 which is not possible, and hence all numbers cannot be imaginary. Alternatively, we can redefine orthogonality for purely imaginary numbers as AA-1 = -I and then the condition becomes identical to real numbers. * For orthogonal 2x2 matrices , out of 4 elements, any three elements should be of one sign, and the rest one of opposite sign. ( for exa, if a,b,c= +ve, d =-Ve ; if a,b,d=-Ve, c=+ve etc) Derivation of condition of Orthogonality for real numbers AAT = I which means :- a2 +b2 =1 c2 +d2 =1 ac+bd=0 then ATA = I which means :- a2 +c2 =1 b2 +d2 =1 ab+cd=0 It follows b=±c now ac=-bd or ab=-bd or a=∓d or when b=+c, a =-d and when b=-c, a=d. A-1 =AT which means :- a=d/(ad-bc)=a/(ad-bc) or ad-bc=+1 when a=d and ad-bc=-1 when a=-d Derivation of condition of Orthogonality for partly real & partly imaginary numbers matrix is A= a ib ic d
AAT = I which means :- a2 -b2 =1 c2 -d2 =1 ac+bd=0 then ATA = I which means :- a2 -c2 =1 d2 -b2 =1 ab+cd=0 It follows b=±c now ac=-bd or ab=-bd or a=∓d or when b=+c, a =-d and when b=-c, a=d. A-1 =AT which means :- a=d/(ad+bc)=a/(ad+bc) or ad+bc=+1 when a=d and ad+bc=-1 when a=-d eigen value λ =(trace/2) ±√ [ (trace/2)2 -Δ ] =a ± √ [ a2 -Δ ] =a ± √ [ a2 -1 ] = a ±b.... for rotation and λ =(trace/2) ±√ [ (trace/2)2 -Δ ] = 0 ±1..... for reflection Ratio of y and x component of eigen vectors (y/x)=(anti-trace/2b) ±√ [ (trace/2)2 -Δ ] /b= -(a/b)±1/b =cosec θ- cotθ or -cosec θ- cotθ |
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circular rotation
S(similarity) matrices (θ) transformation matrix |
circular Reflection S(similarity) matrices(θ) transformation matrix |
Reflection matrices(a) | rotation matrices (a) | |||
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cosθ sinθ
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-sinθ cosθ i -i cosθ -sinθ 1 1 sinθ cosθ i -i -cosθ sinθ 1 1 -sinθ -cosθ i -i -cosθ -sinθ 1 1 sinθ -cosθ i -i sinθ cosθ 1 1 -cosθ sinθ i -i sinθ -cosθ 1 1 cosθ sinθ i -i -sinθ cosθ 1 1 -cosθ -sinθ i -i -sinθ -cosθ 1 1 cosθ -sinθ i -i Similarity transformation matrix S for all the rotation matrices is same & its determinant is -2i. S-1 = 1/2 -i/2 1/2 i/2
-------------------------------------- (non- orthogonal) S (similarity) akin to rotation transformation matrix matrices (θ) ------------------------------------- cosθ sinθ 1 1 sinθ cosθ 1 -1 cosθ -sinθ 1 1 -sinθ cosθ 1 -1 -cosθ sinθ 1 1 sinθ -cosθ 1 -1 -cosθ -sinθ 1 1 -sinθ -cosθ 1 -1 sinθ cosθ 1 1 cosθ sinθ 1 -1 sinθ -cosθ 1 1 -cosθ sinθ 1 -1 -sinθ cosθ 1 1 cosθ -sinθ 1 -1 -sinθ -cosθ 1 1 -cosθ -sinθ 1 -1 The similarity matrix is same for all the above matrices and S-1= 1/2 1/2 1/2 -1/2 ------------------------------------- hyperbolic (non- orthogonal) rotation matrices (θ) -------------------------------------coshθ sinhθ sinhθ coshθ coshθ -sinhθ -sinhθ coshθ -coshθ sinhθ sinhθ -coshθ -coshθ -sinhθ -sinhθ -coshθ sinhθ coshθ coshθ sinhθ sinhθ -coshθ -coshθ sinhθ -sinhθ coshθ coshθ -sinhθ -sinhθ -coshθ -coshθ -sinhθ
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cosθ sinθ
1 1
sinθ -cosθ cosecθ-cotθ -(cosecθ+cotθ) -cosθ sinθ 1 1 sinθ cosθ (cosecθ+cotθ) cotθ-cosecθ cosθ -sinθ 1 1 -sinθ -cosθ (cosecθ+cotθ) cotθ-cosecθ -cosθ -sinθ 1 1 -sinθ cosθ cosecθ-cotθ -(cosecθ+cotθ) sinθ cosθ 1 1 cosθ -sinθ secθ-tanθ -(tanθ+secθ) -sinθ cosθ 1 1 cosθ sinθ tanθ+secθ tanθ-secθ sinθ -cosθ 1 1 -cosθ -sinθ tanθ+secθ tanθ-secθ -sinθ -cosθ 1 1 -cosθ sinθ secθ-tanθ -(tanθ+secθ) There are 4 sets of S each covering 2 reflection matrices. 2sets involve sec and tan and other 2 involve cosec & cot. Determinant for S having sec/tan is -2secθ and that of cosec/cot is -2cosecθ.
(1) S-1 for serial 1,4 is (cosecθ+cotθ)/2cosecθ 1/2cosecθ (cosecθ-cotθ)/2cosecθ -1/2cosecθ (2) S-1 for serial 2,3 is (cosecθ-cotθ)/2cosecθ 1/2cosecθ (cosecθ+cotθ)/2cosecθ -1/2cosecθ (3) S-1 for serial 5,8 is (secθ+tanθ)/2secθ 1/2secθ (secθ-tanθ)/2secθ -1/2secθ (4) S-1 for serial 6,7 is (secθ-tanθ)/2secθ 1/2secθ (secθ+tanθ)/2secθ -1/2secθ ------------------------------------------------------------ (non- orthogonal) S (similarity) akin to reflection transformation matrices (θ) matrix ------------------------------------------- cosθ sinθ -sinθ -cosθ cosθ -sinθ sinθ -cosθ -cosθ sinθ -sinθ cosθ -cosθ -sinθ sinθ cosθ sinθ cosθ -cosθ -sinθ sinθ -cosθ cosθ -sinθ -sinθ cosθ -cosθ sinθ -sinθ -cosθ cosθ sinθ
----------------------------------- hyperbolic (non- orthogonal) reflection matrices(θ) ------------------------------------ coshθ sinhθ -sinhθ -coshθ coshθ -sinhθ sinhθ -coshθ -coshθ sinhθ -sinhθ coshθ -coshθ -sinhθ sinhθ coshθ sinhθ coshθ -coshθ -sinhθ sinhθ -coshθ coshθ -sinhθ -sinhθ coshθ -coshθ sinhθ -sinhθ -coshθ coshθ sinhθ
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a
√ (1- a2 ) √ (1- a2 ) -a -a √ (1- a2 ) √ (1- a2 ) a a -√ (1- a2 ) -√ (1- a2 ) -a -a -√ (1- a2 ) -√ (1- a2 ) a √ (1- a2 ) a a -√ (1- a2 ) -√ (1- a2 ) a a √ (1- a2 ) √ (1- a2 ) - a -a -√ (1- a2 ) -√ (1- a2 ) - a -a √ (1- a2 ) If the difference between the 2 elements of the matrix √ (1- a2 ) and a is x, then x=|√ (1- a2 ) -a | = cosθ - sinθ = |√(1-sin2θ)| and x lies in the range [-√2,√2] and sum of the 2 elements of the matrix √ (1- a2 ) and a is y, then y=|√ (1- a2 ) +a | = cosθ +sinθ = |√(1+sin2θ)| and y lies in the range [-√2,√2] *f(x)=cosx+sinx. cycle is 2Π. amplitude-±√2 x f(x) 0 1 Π / 4 √2 Π / 2 1 3Π / 4 0 Π -1 5Π / 4 -√2 3Π / 2 -1 7Π / 4 0 2Π 1 the dynamic behavior of matrix element a and b where b= √ (1- a2 ) is illustrated below:-
------------------------------------ hyperbolic (non- orthogonal) reflection matrices(θ) -------------------------------------
a √ (a2-1 ) -√ (a2-1 ) -a a -√ (a2-1 )√ (a2-1 ) -a -a √ (a2-1 )-√ (a2-1 ) a -a -√ (a2-1 )√ (a2-1 ) a √ (a2-1 ) a -a -√ (1- a2 ) √ (a2-1 ) -a a -√ (1- a2 ) -√ (a2-1 ) a -a √ (1- a2 ) -√ (a2-1 ) - a a √ (1- a2 )
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a
√ (1- a2 ) -√ (1- a2 ) a a -√ (1- a2 ) √ (1- a2 ) a -a √ (1- a2 ) -√ (1- a2 ) -a -a -√ (1- a2 ) √ (1- a2 ) -a √ (1- a2 ) a -a √ (1- a2 ) √ (1- a2 ) -a a √ (1- a2 ) -√ (1- a2 ) a -a -√ (1- a2 ) -√ (1- a2 ) - a a -√ (1- a2 )
------------------------------------- hyperbolic (non- orthogonal) rotation matrices (θ) -------------------------------------- a √ (a2-1 ) √ (a2-1 ) a a -√ (a2-1 )-√ (a2-1 ) a -a √ (a2-1 )√ (a2-1 ) -a -a -√ (a2-1 )-√ (a2-1 ) -a √ (a2-1 ) a a √ (1- a2 ) √ (a2-1 ) -a -a √ (1- a2 ) -√ (a2-1 ) a a -√ (1- a2 ) -√ (a2-1 ) - a -a -√ (1- a2 )
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Pictorial | Representation of | elements of 2x2 Orthogonal Real | Matrices | |||
rotation matrix with interchange of cos and sin |
rotation /
(grey -anti-clockwise; black-clockwise) matrix |
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sinθ cosθ
-cosθ sinθ (anti-clockwise θ, clockwise π/2) sinθ -cosθ cosθ sinθ (clockwise θ, anti-clockwise π/2) -sinθ cosθ -cosθ -sinθ (anti-clockwise 180-θ, anti-clockwise π/2) -sinθ -cosθ cosθ -sinθ (clockwise 180-θ, clockwise π/2) cosθ sinθ -sinθ cosθ (clockwise θ) cosθ -sinθ sinθ cosθ (anti-clockwise θ) -cosθ sinθ -sinθ -cosθ (clockwise 180-θ)
-cosθ -sinθ sinθ -cosθ (anti-clockwise 180-θ)
Combination no. of combination black-black 2 gray-gray 2 black-gray 2 gray-black 2
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cosθ -sinθ *
0 1 1-2
sinθ cosθ -1 0
cosθ sinθ * 0 -1 2-1 -sinθ cosθ 1 0
-cosθ -sinθ *0 -1 3- 4 sinθ -cosθ 1 0 -cosθ sinθ * 0 1 4-3 -sinθ -cosθ -1 0
-sinθ cosθ * 0 -1 5-7 -cosθ -sinθ 1 0
-sinθ -cosθ * 0 1 6-8 cosθ -sinθ -1 0
sinθ cosθ * 0 1 7-5 -cosθ sinθ -1 0
sinθ -cosθ * 0 -1 8-6 cosθ sinθ 1 0
Tagging 5-7, 7-5 6-8,8-6 1-2, 3-4 2-1, 4-3 |
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Special Matrices | ||||||
* A 2x2 real matrix is
a b c d which is now designed as a b 1/b 1-a such that a is the Golden Number 1. Now determinant Δ =a(1-a) -1 =a -a2 -1= -(a2 -a+1)....(1) tr =1 eigen value = λ =(trace/2) ±√ [ (trace/2)2 -Δ ] = 1/2 ±√ (1/4 -a +a2 +1) = 1/2 ±√ ( a2 -a+5/4) =1/2 ±√ ( a2 -a-1+1+5/4) =1/2 ±√ ( a2 -a-1+9/4) .Taking a2 -a-1 =0, λ =1/2 ±3/2 =2 or -1. Here a =1.618034 and 1-a=-0.618034. λ1+λ2=trace=1 and Δ =-2 Ratio of y , x component of eigen vector (y/x)=(anti-trace/2b) ±√ [ (trace/2)2 -Δ ] /b=(1-2a)/2b ± 3/2b =-(a/b) + (2/b) and -(a/b) +(- 1/b) |
* A 2x2 real matrix is
a b c d which is now designed as a b 1/b 1-a such that the eigen values are the golden numbers. λ1+λ2=trace=1 and λ1λ2=Δ=-1 But Δ =-2-(a2 -a-1)=-1 or a2 -a =0 or a=1 or 0. If b=1, the matrix becomes 1 1 1 0 or 0 1 1 1
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* A 2x2 real matrix is
a b c d which is now designed as a b -1/b 1-a and the eigen values are λ1+λ2=trace=1 and λ1λ2=Δ=-(a2 -a-1) λ=(trace/2) ±√ [ (trace/2)2 -Δ ] = 1/2 ± √[1/4 + (a2 -a-1)]if (a2 -a-1)=0, the matrix is singular and λ= 1/2 ±1/2 if (a2 -a-1)=1, and λ= 1/2 ±√5/2 i.e the golden numbers. if (a2 -a-1)=2, and λ= 1/2 ±3/2 Ratio of y , x component of eigen vector (y/x)=(anti-trace/2b) ±√ [ (trace/2)2 -Δ ] /b=(1-2a)/2b ± 1/2b if Δ=0 =(1/b) - (a/b) +(1/b) and -(a/b) + 0 y/x can also be written as (y/x)=(anti-trace/2b) ±√ [ (anti-trace/2)2 + bc ] /b =(anti-trace/2b) ±√ [ (anti-trace/2b)2 + c/b ] since there is a fuzziness about bc, the numerator of 2nd part of (y/x) remains invariant with respect to inter-change in b,c and also numerical value of b,c so long as product remains the same. when b=c, 2nd part of (y/x) =√ [ (anti-trace/2)2 + b2 ] / b The structure under orange is a Pythagorean triangle. 2nd part of (y/x) = sec ψ or cosec ψ depending on how angle ψ is defined.We define such that 2nd part is sec ψ So (y/x) = tanψ + secψ Let anti-trace/2 =A, b=B=c then, (y/x)=A/B ±√ [ A2 +B2 ] /B =A/B ± H/B where H=√ [ A2 +B2 ] = (A+H)/B =H1/B= tan(ψ+φ). Thus part 2 is made to disappear. See fig.
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Lorentz transformation is a linear transformation. It may include a rotation of space. Rotation free Lorentz transformation is called Lorentz Boost. v is the relative velocity of primed frame with respect the non-primed one. Lorentz Boost Transformation Matrix L (v): γ -βγ = γ2 * 1 --β = γ2 *1 -v/c -βγ γ -β 1 -v/c 1 L-1 : γ βγ βγ γ
Range of β -> ]-1,1[ Range of γ ->[-∞,-1] and [1,∞] Range of βγ is [-∞,+∞] β = v/c ; β2 = v2/c2 γ = 1 / √[1- v2/c2 ] = 1/ √[1-β2 ] γ2= c2 /(c2-v2) β2γ2= v2 /(c2-v2) 1+β2γ2 =γ2 When there are 2 inertial frame of reference , one moving with uniform velocity v with respect to the other , say in x-direction, then if in one frame of reference , the co-ordinate of an object is (x,ct), then the co-ordinates in the other frame (x',ct' ) is given by the transformation equn:- x' =(x ± vt)/√[1- v2/c2 ] t' = (t - vx/c2 ) / √[1- v2/c2 ] or ±γ ∓βγ * x = x' ∓βγ ±γ ct ct' here c is the velocity of light in vacuum. The axes in the moving frame are orthogonal (even though they do not look so). Thus, the Lorentz transformation can be seen as a hyperbolic rotation of coordinates in Minkowski space, where the parameter ϕ represents the hyperbolic angle of rotation, often referred to as rapidity. LT ≠ L-1 , hence L is not part of the orthogonal group. But LTη L= η where η is the minkowski metric. L is a part of the Lorentz group which preserves the quadratic form Σμ,ν 0 to 3 xμηyν for every possible pair of 4-vectors xμ , yν . η = -1 0 0 1 in 2-D space- time. the group of Lorentz transformations is denoted O(3,1), orthogonal with respect to a bilinear form that has signature (3,1); In the second case, the the group of "rotations" is denoted O(3), orthogonal with respect to a bilinear form that has signature (3,0). L(e0) =e0 coshθ +e1 sinhθ L(e1) =e0 sinhθ +e1 coshθ This operator represents a pure boost. It is crucially not symmetric because one of the basis vectors must dot with itself to -1 to represent a hyperbolic geometry. The adjoint is LA(e0) =e0 coshθ -e1 sinhθ LA(e1) =-e0 sinhθ +e1 coshθ It is clear that LLA = a for any vector a. This makes the operator orthogonal. This matrix components may look symmetric. But it is not equal to its adjoint and hence not symmetric in the strictest sense of linear algebra. L' = OL where O is orthogonal, L is pure boost. Above, we have discussed about the boost matrix which does the translation part. However, this can be converted to rotation in hyperbolic space. Let us see how :- L = γ -βγ -βγ γ Range of γ [-∞,-1] and [1,∞] with discontinuity in the range (-1,+1) Range of β is ]-1,1[ βγ = ± 1/√[c2/v2-1 ] Range of βγ is [-∞,+∞] γ2 - γ2β2 =1............(1) Range of γ is similar to coshθ Range of βγ is similar to sinhθ because of (1) as cosh2θ - sinh2θ =1 So L = coshθ sinhθ sinhθ coshθ where θ is a new variable introduced known as rapidity. In this transformation, anti-norm is invariant and not norm. Norm is invariant in circular rotations and anti-norm is invariant in hyperbolic rotations. L-1 = coshθ -sinhθ -sinhθ coshθ L can be expressed as a product of 2 matrices . L= γ 0 * 1 -β 0 γ -β 1 L1 * L2 determinant of L1 is γ2 and that of L2 is 1/ γ2 , so that the product is 1. We can also construct L= cosθ isinθ isinθ cosθ this is also a hyperbolic rotation matrix. In Lorentz transformation, no inertial frame is more privileged than the other and all frames are equivalent. Rapidity/hyperbolic angle that differentiate 2 frames in relative motion , each frame is associated with distance & time co-ordinates . For low speeds, rapidity is proportional to velocity . For higher value of velocity, rapidity increases much faster than velocity and rapidity of light velocity is infinite. For one dimensional motion, rapidities are additive where as velocities are combined as per Einstein's velocity addition formula. Rapidity matrix is of the form p q q p with p2 - q2 = 1 where p,q lie on a unit hyperbola. These type of matrices form indefinite orthogonal group O(1,1) with 1 dimensional Lie Algebra spanned by anti-diagonal unit matrix i.e. 0 -1 -1 0 the indefinite orthogonal group, O(p, q) is the Lie group of all linear transformations of an n-dimensionalreal vector space that leave invariant a nondegenerate, symmetric bilinear form of signature (p, q), where n = p + q. The dimension of the group is n(n − 1)/2. The indefinite special orthogonal group, SO(p, q) is the subgroup of O(p, q) consisting of all elements with determinant 1. Unlike in the definite case, SO(p, q) is not connected – it has 2 components – and there are two additional finite index sub groups, namely l.e. SO.(p,q) and O(p,q)- which has 2 components. Interchanging p with q amounts to replacing the metric by its negative, and so gives the same group. If either p or q equals zero, then the group is isomorphic to the ordinary orthogonal group O(p,q) consists of matrices A such that g-1ATg =A-1 . Here g is a diagonal matrix with diagonal elements (1,1,1,..,-1,-1...). Diagonalizing A gives conjugation of this group with O(p,q) Example of O(1,1) coshθ sinhθ sinhθ coshθ They preserve the quadratic form x2 - y2 and hence preserve thr hyperbola x2 - y2=c where c is a constant The identity component of O(p,q) is often denoted by SO.(p,q) and can be identified with the elements of SO(p,q) that preserve both orientations, This notation is related to the notation O.(1,3) for the orthochronous Lorentz Group. In even dimensions, the middle group O(n,n) is known as the split orthogonal group and is of special interest as it occurs as the group of T-duality transformations in string theory. Split Orthogonal Groups are Chevalley Groups and non-split orthogonal groups are Steinberg groups. Matrices of the form cosz -sinz sinz cosz where z is a complex number are in O(2,C). O(n,C) is not a compact Lie group. The equations defining O(n) in affine space are polynomials of degree 2. Consequently, O(n) is a linear algebraic group.
We know that L * x = x' y y' which means Δx' =γΔx +βγΔy ------(1a) Δy' =βγΔx +γΔy-------(1b) Time dilation : This refers to the relation between time / time difference in rest frame (proper time) and time (improper time) in the inertial frame (moving frame) i.e. a relation between Δy and Δy'. CASE 1;If 2 events occur in same space coordinate in rest frame in different times as reckoned in the rest frame, the same shall be observed in the moving frame as occuring in different times in different locations as governed by equations 1a and 1b. if Δx =0, Δy ≠ 0, then Δy' =γΔy Range of γ ->[-∞,-1] and [1,∞], hence | γ| > = 1 and Δy' > = Δy Hence time as reckoned in the moving frame is dilated or slows down by a factor γ. For example, if a space craft is moving away or towards the earth at a speed of 0.9999c , if an astronaut leaves his chair for 20 minutes as measured in the spacecraft will show a time difference of 24 hours in a earth bound clock. The same effect shall happen for a spacraft observer. That is if an earthly netizen has left his chair for 20 minutes as recorded by the terrestrial clock, the same as appearing in spacecraft clock shall be 24 hours. Hence, Time measurement is an observer dependent notion. CASE 2:If 2 events occur in different space coordinates in rest frame in same times as reckoned at the rest frame (simultaneous events), the same shall be observed in the moving frame as occuring in different times in different locations as governed by equations 1a and 1b. if Δx ≠ 0, Δy = 0, then Δy' =βγ * Δx Range of βγ is [-∞,+∞] , hence | βγ| = [0,+∞] and | Δy' | >= 0, that is events which are simultaneous in the rest frame are not simultaneous in the moving frame. Both are equal only in the limiting case when the frames are in rest with respect to each other in Case 1 as well as Case 2. N.B : One has to explore the relation between Δy' and Δx and Δx' and Δy as well as the sequence of time. Also the impact of uncertainty principle when Δx and Δy become extremely small quantities. To prove that L= secθ tanθ tanθ secθ is equivalent to L= cosθ isinθ isinθ cosθ we first assume that x' = cosθ -sinθ * x y' sinθ cosθ y x' =xsecθ1 + ytanθ1 =xcosθ - ysinθ y' =xx' =xsecθ1 + ytanθ1 =xcosθ - ysinθ =xsinθ + ycosθ Hence x / y =(tanθ1 + sinθ) /(cosθ+secθ1) = (-cosθ+secθ1)/(-tanθ1 sinθ) or cosθ*secθ1 =1 or θ =θ1 . But in the above formulation For Lorentz transformation x'2 - y'2 (in boost ) = x2 - y2 ; But for rotation,
x'2 + y'2 (in rotation ) = x2 + y2 ; To make the rotation matrix compatible with L, we formulate x' = cosθ isinθ * x iy' isinθ cosθ iy which is similar to x' = secθ tanθ * x y' tanθ secθ y Both the forms of L matrix are not orthogonal, though they form a group .
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There is one big difference between Boost matrix
and rotation matrix. cos -sin * x = x' sin cos y y' if one keeps the above rotation matrix R same and interchanges x,y , then R * y ≠ y' x x' This is because transpose of R is not equal to R. RT ≠ R Hence one has to take transpose matrix. So one cannot interchange x,y and keep the equation invariant with same transformation matrix except in special circumstance of angle 45 degree. Whereas in Boost transformation , L * x = x' y y' and L * y = y' x x' because LT = L Hence the equation remains invariant by interchanging x,y with the same transformation matrix. So with hyperbolic metric, interchange of x (space co-ordinate in special relativity) and y coordinate (ct in special relativity) leads to same transformation law. So time and space are treated at par in special relativity. comparison between Rotation & Boost transformation: R (θ ) *x = x' y y' L(hθ) * x = x' y= y' Putting the values of the matrices, we get x'2 + y'2 (in rotation ) = x2 + y2 ; x'2 - y'2 (in boost ) = x2 - y2 ; x'2- y'2 (in rotation ) =(x2 - y2 )cos2θ -2xysin2θ x'2+y'2(in boost )=(x2 + y2 )cosh2θ +2xysinh2θ sinh(90) = i ; cosh(90)=0; so at θ =45° ,x'2- y'2 (in rotation ) =(-1)* x'2+y'2(in boost ) Conservation Laws : When space-time has a Killing Vector, geodesics have a constant value of vbζb where vb is velocity of four vector. For example, because the Schwartzchild metric has a Killing vector ζb = ∂t , test particles have a constant value of vt and therefore momentum pt is conserved interpreted as mass-energy. A flat 1+1 dimensional space-time has Killing Vectors ∂x ,∂t .Corresponding to these are conservation of momentum p and mass-energy E. If we make a Lorentz boost, these 2 Killing vectors get mixed by a linear transformation of p,E in a new frame. If any set of points is displaced by where all distance relationships are unchanged (i.e., there is an isometry), then the vector field is called a Killing vector. *In mathematics, a Killing vector field (often just Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point on an object the same distance in the direction of the Killing vector field will not distort distances on the object. Definition. A space-time is a pair ( M, g ), where M is a four dimensional differentiable manifold and g is a metric of the Lorentzian signature (+ − −− ) given on it. *Killing vector fields on a Lorentzian manifold are oriented time-like, light-like, or space-like. *A homogeneous and isotropic n-dimensional manifold is necessarily a constant-curvature space with maximal number n(n + 1)/2 of linearly independent Killing vector fields We can rewrite Lorentz boost transformation as L : γ βγ βγ γ since β and γ are pure ratios , we can write β = sinθ γ =secθ βγ =tanθ
L = secθ tanθ tanθ secθ All the L , L-1 matrices are commutative under matrix multiplication,L =LT, have determinant +1,anti-norm of row/col vectors is 1 ,anti-dot product is zero, signature is ++++, ----, ++-- . These matrices form an abelian group under matrix multiplication secθ tanθ * secφ tanφ = tanθ secθ tanφ secφ secθsecφ+tanθtanφ secθtanφ+tanθsecφ = secθtanφ+tanθsecφ secθsecφ+tanθtanφ A B B A where A = (1+sinθsinφ)/cosθcosφ B = (sinθ+sinφ)/cosθcosφ A2 -B2 =( 1/cos2θcos2φ) * (1+sin2θsin2φ-sin2θ-sin2φ) = ( cos2θ-cos2θsin2φ )/( 1/cos2θcos2φ) = cos2θcos2φ / cos2θcos2φ =1 So A,B can be put as A =secη B =tanη Since the eigen values of matrix L are ( secθ + tanθ ) & ( secθ - tanθ ) , L can be diagonalized to Ld = secθ + tanθ 0 = 0 secθ - tanθ
(cosθ/2 +sinθ/2)/(cosθ/2 -sinθ/2) 0 0 (cosθ/2 -sinθ/2)/(cosθ/2 +sinθ/2) =Ld1 *Ld2 where Ld1=cosθ/2 +sinθ/2 0 0 cosθ/2 -sinθ/2 = x1 0 0 y1
Ld2= 1/(cosθ/2 -sinθ/2) 0 0 1/(cosθ/2 + sinθ/2) = 1/y1 0 0 1/x1 L-1d = secθ - tanθ 0 = 0 secθ + tanθ
(cosθ/2 -sinθ/2)/(cosθ/2+sinθ/2) 0 0 (cosθ/2 +sinθ/2)/(cosθ/2 -sinθ/2) =L-1d1 *L-1d2 where L-1d1=cosθ/2 -sinθ/2 0 0 cosθ/2 +sinθ/2
L-1d2= 1/(cosθ/2 +sinθ/2) 0 0 1/(cosθ/2 - sinθ/2) Range of cosθ/2 ± sinθ/2: cosθ/2 ± sinθ/2 = √2( cosθ/2 *1/√2 ±sinθ/2 *1/√2 ) = √2( cosθ/2 *sin 45° ± sinθ/2 *cos45°) = √2 sin (45° ± θ/2) Hence Range of cosθ/2 ± sinθ/2 is [-√2 ,√2 ] since range of sin of an angle is [-1.1] . Now x1 + y1 =2cos(θ/2) x1 - y1 = 2sin(θ/2) x1y1 = cosθ (1/y1)*(1/x1) =secθ As cosθ --> 0 secθ--->∞ in such a way that cosθ*secθ =1 or d/dθ (cosθ*secθ) =-tanθ +tanθ=0 Here slopes are axis independent of orientations in space or time . All that is required is two axis should be orthogonal. We normalize x1, y1 so that x1=(cosθ/2 +sinθ/2)/√2 y1=(cosθ/2 -sinθ/2)/√2 x2= √2 / (cosθ/2 -sinθ/2) y2= √2 / (cosθ/2 +sinθ/2) Finding a separate frame of Reference : We know that the Lorentz transformation matrix L transforms (x,y) into (x',y') where both (x,y) and (x',y') lie on the same hyperbolic trajectory and the anti norm of the state vectors remains invariant under such transformations. x' = secθ tanθ * x y' tanθ secθ y or x' =xsecθ + ytanθ ......(1) y'=xtanθ + ysecθ........(2) Now L* =tanθ secθ secθ tanθ is also a Lorentz transformation matrix and y' =xsecθ + ytanθ.......(3) x'=xtanθ + ysecθ .......(4) Difference between L and L* is that in (1) transformed co-ordinate is x' where as in (3) , RHS remaining the same, the transformed coordinate is y'. same is true for (2) and (4). Moreover determinant is -1 in L* . Physically, if we put x as space axis and y=ct as restructured time axis, L transforms (x,ct) to (x',ct') and the new frame of reference is an inertial frame. But under L*, (x,ct) is transformed into (ct',x') which indicates that space coordinate becomes time coordinate and vice versa. What sort of frame of reference such transformation signify ?? L L*-1 = σ1 . (pauli matrix) L*iL = σ3 iL = isecθ itanθ = itanθ isecθ -1 * secθ tanθ = tanθ secθ secθ tanθ -tanθ -secθ However in order to produce σ2 ,The matrices have to be hermitian/anti hermitian. secθ itanθ * -itanθ secθ tanθ -isecθ = isecθ tanθ 0 -i = σ2 . i 0 We also find that secθ itanθ * -itanθ secθ secθ itanθ * -itanθ secθ =sec2θ+tan2θ isecθtanθ = -isecθtanθ sec2θ+tan2θ secη itanη -itanη secη L2 = sec2θ+tan2θ 2secθtanθ = 2secθtanθ sec2θ+tan2θ secη tanη tanη secη where secη =sec2θ+tan2θ or cosη =(1-sin2θ) /(1+sin2θ) and sinη =(2sinθ) /(1+sin2θ) hence ST ratio=sinη /sinθ = 2 /(1+sin2θ) ST (min) =1 ST (max)=2 |
Comparison
Orthogonal Rotation Matrix & a b = ±x ∓√(1-x2) c d ±√(1-x2) ±x (1) the sign of 4 matrix elements are such that 3 are of one sign, and the other is of opposite sign. Exa- +++-, ---+, +-++, -+--, etc (2) Δ =1 Δ = ad-bc= x 2 + y2 =1 the matrix is the function of a single variable x and y= ±√(1-x2) (3) a2 + b 2 = a2 + c 2 =1 (4) ab=-cd , ac=-bd (5) Δ =1 implies that a=d and b=-c (6) tr = 2x or -2x with range [-2,2] (7) AT =A-1 ; (8) a2 + b 2 = a2 + c 2 =1 implies that cos2θ +sin2θ =1 if we put a =cosθ and b =sinθ (9) Rotation is in circular Euclidian metric. (10) This is the transformation matrix for active and passive rotation. (11) this is an orthogonal matrix in circular case. (12) norm remains invariant |
between Non-Orthogonal matrix akin to Rotation a b = ±x ±√(x2-1) c d ±√(x2-1) ±x (1) the sign of 4 matrix elements are such that either (a) all 4 matrix elements are of one sign, such as ++++,---- (b) 2 are of one sign and the other 2 of opposite sign, such as ++--,+--+, -+-+ etc. (2) Δ =1 Δ =ad-bc=x 2 - y2 =1 the matrix is the function of a single variable x and y= ±√(x2-1) (3) a2 - b 2 = a2 - c 2 =1 (4) ab=cd , ac=bd (5) Δ =1 implies that a=d and b=c (6) tr = 2x or -2x with range ]-2,2[ (7) AT =A ; (8) a2 - b 2 = a2 - c 2 =1 implies that cosh2θ -sinh2θ =1 if we put a =coshθ and b =sinhθ (9) Rotation can be construed in hyperbolic Euclidian metric. (10) This is the transformation matrix for Lorentz boost transformation where one inertial frame moves in x-direction with respect to another inertial frame at a constant velocity v . This is a linear inertial frame transformation which is mathematically equivalent to a hyperbolic rotation. (11) this can be construed as an orthogonal matrix in hyperbolic case. (12) anti-norm remains invariant or norm remains invariant if we define norm as x2 - y 2 =1 in hyperbolic case. |
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In rectangular hyperbola (xy=r2), the transformation matrix is Ad= a 0 0 ±r2/a where r is a number. x' = Ad * x y' y Now P Ad P-1 =A where P is orthogonal and is the transformation matrix. The inverse curve of a rectangular hyperbola with inversion center at the center of the hyperbola is a lemniscate (Wells 1991). If 3 vertices of a triangle lie on a rectangular hyperbola, then the orthocenter also lies on the hyperbola.Equivalently, if four points form an orthocentric system, then there is a family of rectangular hyperbolas through the points. Moreover, the locus of centers of these hyperbolas is the nine-point circle of the triangle (Wells 1991).If four points do not form an orthocentric system, then there is a unique rectangular hyperbola passing through them, and its center is given by the intersection of the nine-point circles of the points taken three at a time (Wells 1991). Conversion of normal hyperbola to Rectangular Hyperbola : x2 /a2 - y2 /b2 =1 ; If a=b, x2- y2 = a2 ; We have some rectangular hyperbola of the form xy= a2/2 In order to make this hyperbola open horizontally , we need to rotate it 45 degree clockwise . We now express x,y in terms of new coordinates u, v as x=(1/√2)u - (1/√2)v y=x=(1/√2)u + (1/√2)v Substituting this in xy= a2/2 , we get u2- v2 = a2 ; which is analogous to x2- y2 = a2 ; This tells us that our originally diagonally opening rectangular hyperbola is a rotation by 45 degree of the horizontally opening hyperbola of the form x2- y2 = a2 ; Conversion of unit hyperbola to unit Circle : x2- y2 = 12 = x2+(iy)2 which is a circle with one axis real and another imaginary. Now x' = cosθ -sinθ * x or x' = cosθ -isinθ * x iy' sinθ cosθ iy y' -isinθ cosθ y Effecting rotation of 45 degree (active/passive) x' = (1/√2)(x-iy) iy' = (1/√2)(x+iy) and x2+(iy)2 =x'2+(iy')2 =1 => vector norm is preserved.
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Upper Triangular Matrix : a b 0 d Eigen Value- a, d (y/x) of eigen vectors: [(d-a)/(2b)]*3 and [(d-a)/(2b)]*1 If a=d, (y/x) =0 |
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Special Relativity : If Σ is unprimed frame of reference and Σ' is the primed frame moving at uniform speed of v in positive x-direction, then as per Newtonian Mechanics, x'= x-vt .......(1) x=x'+vt' ......(2) In terms of transformation matrix, we can put 1 -v * x = x' 0 1 t t where matrix R = 1 -v 0 1 is a invertible, upper triangular matrix. Now R-1 = 1 v 0 1 and 1 V * x' = x 0 1 t' t' For R, eigen values are 1 and (y/x) =0 In special relativity, t' is not equal to t . It has 2 main postulates- That the laws of physics are the same in all inertial frame of reference. AND The velocity of light in vacuum is a constant irrespective of the velocity of its source. In Newtonian mechanics, transmission of information regarding the occurrence of an event in primed frame to unprimed frame in instantaneous, hence t'=t. Not so in special relativity as it is transmitted at the best in the speed of light i.e. c. the relative velocity of light signal fromΣ' to Σ is (c+v) and time taken to traverse is (x-vt)/(c+v). By this time light travels a distance of c*(x-vt)/(c+v). Similarly, if the signal travels from Σ frame to Σ' frame, relative velocity is (c-v) , time taken is (x-vt)/(c-v) and distance traversed is c*(x-vt)/(c-v). Since both the expressions have to be equivalent, the distance is geometric average of the two distances i.e. √ (distance1 *distance2) or x'=√[c2*(x-vt)2 / (c -v )2 ]= (x-vt)*γ where γ= √[1/(1-c2/v2)] or x' /γ = x-vt ......(3) similarly x/γ = x'+vt' ......(4) Eliminating x', from (3),(4), we get t' = (t- vx/ c2 )*γ and transformation law is L * x = x' ct ct' Determinant of L=1, anti-trace=0; Eigen Value of L matrix: λ1=√[(1+β)/(1-β)] =√[(c+v)/(c-v)] λ2=√[(1-β)/(1+β)] =√[(c-v)/(c+v)] (y/x) of eigen vectors = 0 ±1 i.e. the ratio is independent of velocity v. Norm is 2x2 and anti-norm is zero.. |
Hyperbolic Functions: domain range sinh(x) [-∞,+∞] [-∞,+∞] cosh(x) [-∞,+∞] [1, +∞]
tanh(x) [-∞,+∞] [-1,+1 ] coth(x) [0, +∞] [1, +∞] sech(x) [-∞,+∞] [0, +1] cosh2(x) - sinh2(x) =1 coth2(x) - cosech2(x) =1 sech2(x) + tanh2(x) =1 sinh(2x) =2sinh(x)cosh(x) cosh(2x)=cosh2(x) + sinh2(x) tanh(2x)=2tanh(x)/(1+tanh2(x)) sinh(x)=(ex - e-x) / 2 cosh(x)=(ex + e-x) / 2 ex ≠ 0 ∀ x ∈ R . Hence ex is continuous over the entire range of real numbers. Since e-x is given by e-x = 1/ex , it is also continuous in R. coshx+sinhx=ex coshx + sin hx has domain [-∞ ,+∞ ], Range [0,+∞] coshx - sin hx has domain [-∞ ,+∞ ], Range [+∞, 0] coshx-sinhx= e-x * sinh(x) is neither bounded above nor below over the whole of R and hence has no minimum or maximum. But over every bounded interval, it has minimum and maximum value * hyperbolic functions take a real argument called hyperbolic angle. The size of the hyperbolic angle is twice the area of hyperbolic sector. *sinh(2x)=2sinhx*coshx cosh(2x)=cosh2 x +sinh2x |
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* Any square matrix can be converted into either of the following:- a) rotation matrix b) reflection matrix c) non-orthogonal matrix akin to rotation d) non-orthogonal matrix akin to reflection Procedure: find out a*d. take absolute value of ad and then find out positive square root, say ad. If a is positive, write + square root √ad. If a is negative , write a=-√ad. This is the average of a,d. Similarly, write for d , taking positive or negative square root i.e. √ad depending on whether d is + or -. Follow similar procedure for b,c and replace them by +√bc or -√bc depending on whether b/c was positive or negative. Then divide all matrix elements by positive square root of determinant . It will be either c or d as above depending on the resultant determinant being +1 or -1. Exa- 16 -18 2 4 signature is +++- determinant= 100 √bc =6 , since b is -ve, b=-6 c is positive ,so c=6 similarly √ad =8 a=8 b=8 the matrix is 8/10 6/10 6/10 8/10 determinant=+1 signature is ++++ . a=d, b=-c, hence rotation matrix. Exa- 16 18 2 4 determinant= 28 Signature is ++++. √bc =6 , since b is +ve, b=6 c is positive ,so c=6 similarly √ad =8 a=8 b=8 the matrix is 8/√28 6/√28 6/√28 8/√28 determinant=+1 Hence the matrix is c type. here AT = A Had the determinant been -1, it would have been d type and A-1 =A.
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Lorentz Boost transformation
|v| =0 | |v| =c/√2 =0.7072c | |v| =c | |v| =√2 c=1.414c | |v| = ∞ | ||
|γ| |
1 / √[1- (v2/c2)] | 1 | √2=1.414 | ∞ | i*1 | i* 0 |
|βγ| |
1 / √[(c2/v2)-1] | 0 | 1 | ∞ | i*√2 | i*1 |
|βγ| / |γ| =|β| |
v/c | 0 | 1 / √2 =0.7072 | 1 | √2 | ∞ |
|γ|2 - |βγ|2 | 1-0=1 | 2-1=1 | ∞ - ∞ =1 | -1+2 =1 | ||
*When |v| < = |c|, |γ| is
forbidden to have value (0,1) but |βγ|
can have this value *when |v| ∈ [0, c/√2], |v| ∈ [∞, √2c], |γ| ∈ [1,√2] |γ| ∈ [0,i*1] |βγ| ∈ [0,1] |βγ| ∈ [i*1, i*√2] The range of red colored and violet colored ratios are same except for the fact in one set the axis are imaginary and |γ| and |βγ| swap ranges. |