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)

 
  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

 
  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:-yx

Eigen vector(y/x):1

ev1=i' +j'

 

ev2=i' - j'

<ev1,ev2> =1-1=0

eigen value:-yx

Eigen vector(y/x):1

ev1=i' +j'

ev2=i' - j'

<ev1,ev2> =1-1=0

  eigen value:-yix

Eigen vector(y/x):i

ev1=i' +i*j'

ev2=i' -i*j'

<ev1,ev2> =1-1=0

eigen value:-yix

Eigen vector(y/x):i

ev1=i' +i*j'

ev2=i' -i*j'

<ev1,ev2> =1-1=0

 
             
             
             
  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.

 
  ∓√(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 = e

cosθ -i sinθ = e-iθ

-cosθ +i sinθ = -e-iθ

-cosθ -i sinθ = -e

If x=sinθ , eigen values are

sinθ +i cosθ = ie-iθ

sinθ -i cosθ = -ie

-sinθ +i cosθ = ie

-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] ) =√(1sin2θ ) 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

 
  circular rotation    S(similarity)

matrices (θ)          transformation matrix

circular Reflection   S(similarity)

 matrices(θ)           transformation matrix

  Reflection matrices(a) rotation matrices (a)  
1

 

2

 

3

 

4

 

5

 

6

 

7

 

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.

 

2.

 

3.

 

4.

 

5.

 

6.

 

7.

 

8.

 

 

 

 

 

 

 

 

 

1.

 

2.

 

3.

 

4.

 

5.

 

6.

 

7.

 

8.

cosθ   sinθ               1      1         

-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

 

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

 

 

 

 

 

1

 

2

 

3

 

4

 

5

 

6

 

 

7

 

8

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 )

 

 

 

 

 

 

 

 

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 )

 

 

 

 

 

 

 

 
 

 
  Pictorial Representation of    elements of   2x2 Orthogonal Real Matrices  
 

 
 

rotation  matrix with interchange of cos and sin

rotation /              (grey -anti-clockwise; black-clockwise)

 matrix                       

       
1

 

 

2

 

 

3.

 

 

4

 

5.

 

 

6

 

 

7

 

 

 

8

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

 

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

       
  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

 

 

 
  * 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.

   
 

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+β2γ22

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

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

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() * 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 )

 

  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.

 
  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

         
  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

       
 

* 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.