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[ ]       |
|       +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† =H†H = H2 * All Hermitian matrices are normal since HH† =H†H  . 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† =H†H =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.[σi]  *  [σj]  +  [σj]  *  [σi]      ={σi,σj}= 2δijI  and [σi,σj]= 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+=σ3 ψy- ψz+=σ1 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-z2Hence 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 beP.σ =   z          x+iy       -(x-iy)        z  and Pauli matrices will berσ1= 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 berσ1'= 0   -1   rσ2'= 0  -i    rσ3'= -1  0    each having determinant +1.        1    0           -i    0            0   -1Hence it is all the same whether we take (rσ1,rσ2 ,rσ3 ) or (rσ1',rσ2' ,rσ3' ) as the set of Pauli matrices each having determinant 1. Transformation matrix  for Lorentz boost is given byL = 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 {I, iσ1, iσ2, iσ3} is isomorphic to the real algebra of quaternions ℍ. The isomorphism from ℍ 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 ->iσ1So 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 ascos θ/2    i sin θ/2  =  cosθ/2*I + iσ1sinθ/2  where σ1= 0   1  and I = 1  0i 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,