MATRICE : H --(2x2)

   Det |H|=+i

 

+i

+i

 

 

  X
     Det |H1|=+i

 

+i

+i

 

 =
     Det |H2|=+i

+i

 

+i  

+i

 

 +i

Det |C|=+i[]   Det |C1|=+i[]    
co-factors:   co-factors:    
                  |       +i[]                     |       +i[]    
        +i[ ]  |        +i[ ]       |         
Adjoint: Det |A|=+i[]   Adjoint: Det |A1|=+i[]    
                  |       +i[]                     |       +i[]    
+i[ ]       |        +i[ ]       |         
Inverse: Det |H-1|=+i[]   Inverse: Det |H1-1|=+i[]    
+i                  |       +i[]   +i                  |       +i[]    
+i[ ]    |      +i  []   +i[ ]   |      +i  []    
(Det |H|)*Det |H-1|=+i   (Det |H1|)*Det |H1-1|=+i    
         
Whether matrix is Hermitian ?   Whether matrix is Hermitian ?    
Put Arbitrary value of 2nd component of Eigen Vector:

b =  +i 

  Put Arbitrary value of 2nd component of Eigen Vector:

b =  +i

   
Ratio of eigen vectors for λ1(a1:b)                               
real part--         :                                               
Imaginary part-- :    

  

       
 Ratio of eigen vectors for λ2(a2:b)                               
real part--         :                                               
Imaginary part-- :    
       
1st Eigen vector component for λ1(a1):+i()   1st Eigen vector component for λ1(a1):+i()    
1st Eigen vector component for λ2(a2):+i()   1st Eigen vector component for λ2(a2):+i()    
1st Eigenvector component normalized for λ1:        
1st Eigenvector component normalized for λ2:        
       
Eigen Equn:  λ2 + λ[]+[]=0   Eigen Equn:λ2 + λ[]+[]=0    
     
λ1=  + i()*    λ1=+ i()*    
λ2=+ i()*    λ2=+i()*    
Below is   H*H   matrix   Since H =H, Det |HH  | =

If Det |HH  | = 1,H  is a unitary matrix & H = H-1

 

+ i

+i

 

 

* If H is invertible, then 1/ λ  is an eigenvalue of H-1   . If k is any number, λ + k is an eigenvalue of H +kI
* If A is a square matrix, then  This theorem is known as Caley-Hamilton theorem.
* λ  are the eigen values. 2-D space vectors/spinors will have 2 eigen values. λ =0 implies matrix is singular. * If  λ is imaginary/complex, then the matrix does not have an eigenvalue & hence no eigenvector.  The dimension of the eigen vector corresponding to an eigenvalue is less than or equal to the multiplicity of that eigenvalue.
* Hermitian matrices (H) are square matrices with complex entries in off-diagonal elements. aij =(aji)* . The real analog is symmetric matrices. H is called Hermitian conjugate of H and H=H. HH is also hermitian and

HH =HH = H2

* All Hermitian matrices are normal since HH =HH  . The real counterpart is that  product of real symmetric matrices with their transpose is commutative and hence normal.  HTH =HHT
* A hermitian matrix H is called Unitary if  HH =HH =I where I is the Identity matrix. In that case H=H =H-1   .
* Well known 2x2 Hermitian matrices are Pauli Matrices named after the renowned physicist Wolf Gong Pauli. These are unitary also. There are 4 Pauli matrices including the identity matrix. Out of these, 3 have all real elements and one has complex numbers.

* The commutation relation of Pauli matrices are

* We define anti-commutator as
* [σi]2=I where i=1,2,3.

i]  *  [σj]  +  [σj]  *  [σi]      ={σij}= 2δijI  and [σij]= 2iεijkσk

Pauli eigen vectors are

ψx+=(1/√2 ) 1

                     1

ψx-=(1/√2 ) 1

                  -1

 

ψy+=(1/√2 ) 1

                    i

ψy-=(1/√2 ) 1

                  -i

ψz+ =         1

                  0

ψz- =          0

                  1

and

ψx+=iσ2 ψx-  

ψy+3 ψy-

ψz+1 ψz-

 

ψx-=-iσ2 ψx+  

ψy-=  σ3 ψy+

ψz-=  σ1 ψz+

ψy+=iσ1 ψx+

Hence there can be transformation from ψx  to ψy    state and vice versa but not to ψz    state.

* The Pauli matrices and the 2x2 Identity matrice I form a complete set. Any 2x2 Hermitian matrix, say A can be expressed as a linear combination of any 3 of these 4 matrices

   A =C0σ0  +C1σ1 + C2σ2  + C3σ3      where   C0, C1 ,C2 ,C3  are the linear co-efficient and are represented by real numbers. A 3-D Pauli vector is represented as

  σ= σ1x‾ + σ2y‾ +σ3 z‾ , x‾ , y‾ ,z‾ being basis vectors in x,y,z axis respectively. If P is a point in 3-D Eucledean space with co-ordinates (x,y,z)

 then vector P = xx‾+yy‾+zz‾ where x‾ , y‾ ,z‾ being basis vectors in x,y,z axis respectively. Now P.σ =xσ1 + yσ2+zσ3 ;which is a matrix representation of vector P & the determinant of this matrix is the negative norm square of the vector . Thus we have mapping from vector basis to the Pauli matrix basis for representing a 3-D vector . Now P.σ = z        x-iy

                             x+iy      -z  and determinant = -x2 -y2-z2

where x,y,z are co-ordinates of point P. This matrix is a Hermitian matrix with zero trace.

If we take σ1'=0  -1   σ2'=0  i   σ3'=-1 0 

                           -1  0       -i  0         0 1

then 

P.σ = -z        -x+iy

       -x-iy        z        and determinant = -x2 -y2-z2

Hence it is all the same whether we take (σ1,σ2 ,σ3 ) or (σ1',σ2' ,σ3' ) as the set of Pauli matrices each having determinant -1. .

If we want make the determinant x2 +y2+z2  , the matrix will be

P.σ =   z          x+iy

       -(x-iy)        z 

and Pauli matrices will be

1= 0   1   rσ2= 0  i    rσ3= 1  0    each having determinant +1.

        -1  0           i   0            0   1

Alternatively,

P.σ =   -z          -(x+iy)

       (x-iy)         -z 

and Pauli matrices will be

1'= 0   -1   rσ2'= 0  -i    rσ3'= -1  0    each having determinant +1.

        1    0           -i    0            0   -1

Hence it is all the same whether we take (1,2 ,3 ) or (1',2' ,3' ) as the set of Pauli matrices each having determinant 1.

Transformation matrix  for Lorentz boost is given by

L = cosθ    i sinθ  =  cosθ*I + iσ1sinθ  where σ1= 0   1  and I = 1  0

      i sinθ    cosθ                                                     1   0               0  1

* We know that for spin 1/2 particles like electrons, these must be rotated by an angle 4π radian in order to return to their original configuration, since these rotations are not in 2-D but in 3-D space ( dimensionality matters ) which means there is a 2 to 1 correspondence between su(2) and so(3). Hence for rotation θ  for so(3), su(2) rotation will be θ/2. Lie algebra su(2) is isomorphic to lie algebra so(3) which corresponds to Lie Group SO(3).For spin 1/2 particles, the spin operator is given by J=(h/2)σ which is a fundamental representation of SU(2).

*The real linear span of {I123} is isomorphic to the real algebra of quaternions . The isomorphism from  to this set is given by the following map (notice the reversed signs for the Pauli matrices):

  1 -> I , i ->-iσ1   ,j ->-iσ2 , k ->-iσ3

Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,

1 -> I , i ->iσ3   ,j ->iσ2 , k ->iσ1

So transformation matrix Q for rotation of spin 1/2 particles about x-axis by an angle θ may be written in terms of Pauli Matrices as

cos θ/2    i sin θ/2  =  cosθ/2*I + iσ1sinθ/2  where σ1= 0   1  and I = 1  0

i sin θ/2    cos θ/2                                                           1   0               0  1

* If n is a unit vector (in any orientation) , then n=xx‾ +yy‾ +zz‾  and   x2 +y2+z2=1, and

  n.σ = -z          -(x+iy)  and (n.σ)2 =x2 +y2+z2          0         =   1      0  = I or (n.σ)2p =I where p is any integer and  (n.σ)2p+1 = n.σ

           (x-iy)         -z                              0              x2 +y2+z2      0      1

      

*Dimension of the Hilbert space corresponding to spin state s is 2s+1. Elements of this vector space are called spinors. We have seen that

   &   are eigen states of Sz operator and these spinors form a basis for 2-D spinor space. For σx  , the eigen values are =  and λ=+ 1 and -1.

For  λ=+ 1 , the eigenvector equation is which gives a=b. To normalize the eigen state,    so that the state is 

For λ=- 1 , the eigenvector equation is  which gives a=-b. And the normalized eigen state is .   Similarly for σy , we get  2 normalized eigen states as for  λ= 1 and -1 respectively. This can be written in terms of σz  eigen states  as  . Since the coefficients of the two σz  eigen states are equal in magnitude, if σz  is measured for a particle in σy   eigen state, it is equally likely to be spin up or down.

Rotation matrix are :-    and   . The 2x2 rotation matrix must have the following form-- with a^2 + b^2 =1
*Those matrices where A = A-1  , are called involutary matrices.
* If A is a matrix, B is a matrix , then AB may be a null matrix under certain circumstances. What are the circumstances ? ( For 2x2 matrices)

* Matrix A is called IDEMPOTENT Matrix if A2 =A. Identity matrix is the only IDEMpotent matrix having a non-zero determinant.

Link-1,2,3,4,5,6,7,8,9,10,