* 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 CaleyHamilton theorem. 
* λ are the eigen values. 2D 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 offdiagonal 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
= H^{2} 
* 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.
H^{T}H =HH^{T} 
* 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 anticommutator as

* [σ_{i}]^{2}=I where i=1,2,3. [σ_{i}]
* [σ_{j}]^{ } + [σ_{j}]
* [σ_{i}] = 2δ_{ij}I 
* 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 =C_{0}σ_{0} +C_{1}σ_{1}
+ C_{2}σ_{2} + C_{3}σ_{3}
where C_{0}, C_{1}
,C_{2} ,C_{3} are the linear coefficient. 
*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 S_{z} operator and these spinors form a
basis for 2D spinor space. For σ_{x}
, 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 σ_{y} , we
get 2 normalized eigen states as
for
λ= 1 and 1 respectively. This can be written in terms of σ_{z}
eigen states as
. Since
the coefficients of the two σ_{z}
eigen states are equal in magnitude, if σ_{z}
is measured for a particle in σ_{y}
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 

Link1,2,3,4,5,6,7,8,9,10, 