MATRICE : H --(2x2)

   Det |H|=+i

 

+i

+i

 

 

  X
     Det |H1|=+i

 

+i

+i

 

 =
     Det |H2|=+i

+i

 

+i  

+i

 

 +i

co-factors: Det |C|=+i[]   co-factors: Det |C1|=+i[]    
                  |       +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]      = 2δijI

* 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 these 4 matrices

   A =C0σ0  +C1σ1 + C2σ2  + C3σ3      where   C0, C1 ,C2 ,C3  are the linear co-efficient.

*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
 
Link-1,2,3,4,5,6,7,8,9,10,