Linear Operators and Matrices
Now, see a linear operator is just a matrix. Suppose and are basis of and is a basis of then,
Ok so imagine an operation such that the following shit holds ok?
In finite dimensions, inner product space i.e., vector spaces equipped with inner prducts for all vector space Hilbert Space
Consider to be orthonormal basis, we have —
Norm of a vector
We can say that is normalized iff .
A set of vectors is orthonormal if i.e., and .
Gram Schmidt: for orthonormal basis
From this notion we obtain the completeness relation, .
A Hilbert Space is complete which means that every Cauchy sequence of vectors admits in the space itself. Under this hypothesis there exist Hilbert bases also known as complete orthonormal systems of vectors in .
For any orthonormal basis of , we have the following.
Eigenvectors and Eigenvalues
Under a given linear transformation , where s.t. they do not get shifted off their span.
All such vectors are referred as eigenvectors and gives all possible eigenvalue.
It is the space of all vectors with a given eigenvalue . When an eigenspace is more than one dimensional, we call it degenerate.
Adjoints and Hermitian
Suppose then such that we have,
This operator is called as the adjoint or Hermitian conjugate of the operator .
A normal matrix is Hermitian if and only if it has real eigenvalues.
If then is positive semi-definite and has positive eigenvalues.
If a Hermitian matrix has positive eigenvalues then it is positive semi-definite.
If then it is both Hermitian and positive semi-definite.
All positive semi-definite operators are Hermitian, by definition.
diagonalizable if and only if it is normal.: A linear operator is
Some notes and derivation regarding the above:
where 's are linearly independent eigenvalues of .
Matrices and Vectors
In the following statements we are dealing with as a orthonormal basis set.
Now, to represent a operator or linear transformation as matrix in orthonormal basis.
Now diagonalization for any normal matrix.
where are eigenvalues of under a given orthonormal basis set for vector space , each is an eigenvector of with eigenvalue .
If is Hermitian, all eigenvalues are non-negative.
forms the linear operator that acts on vector space givern that acts on and acts on .
Properties of trace are given below as follows.
for orthonormal basis
The above properties yield certain implications as follows.
, with algebraic multiplicities
Entanglement excludes the possibility of associating state vectors with individual subsystems. Therefore, we introduce density matrices and the corresponding idea of reduction preformed with partial trace.
Hilbert-Schimdt Inner Product
forms the vector space of operators over the Hilbert space . Then, we can show that is also a Hilbert space with as the inner product operator on .
Also, we have .
Commutator and Anti-commutator
Theorem of Simultaneous Diagonalization
Suppose and are both Hermitian matrices, then iff orthonormal basis such that both and are diagonal with respect to that basis.
Polar Value Decomposition
If is any linear operator and is a unitary then are positive operators, such that
Moreover, if exists, then is unique.
Singular Value Decomposition
If is a square matrix and unitaries then is a diagonal matrix, such that
where has non-negative values.
has non-negative eigenvalues then, is possible where has non-negative values.: If