|* 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.
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
|* 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.
|* A hermitian matrix H is called Unitary if HH† =H†H
=I where I is the Identity matrix. In that case H=H†
|* 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.
* [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
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
gives a=b. To normalize the eigen state,
so that the state is
For λ=- 1
, the eigenvector equation is
gives a=-b. And the normalized eigen state is
Similarly for σy , we
get 2 normalized eigen states as
λ= 1 and -1 respectively. This can be written in terms of σz
eigen states as
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 :-
The 2x2 rotation matrix must have the following form--
with a^2 + b^2 =1