hermitian adjoint

In mathematics, the Hermitian adjoint of an linear operator is a matching operator (very similar to the inverse operator in concept) defined over a linear space with inner product. It is also called Hermitian conjugate or just adjoint. When talking about matrices it is also called the conjugate transpose.

In Hilbert space

The Hermitian adjoint can be generally defined using only the axioms (properties) of Hilbert space and the inner product.

Definition

For every linear operator

A

(which is bounded; see operator's norm), the Hermitian adjoint, marked by

A^\dagger

(pronounced "A dagger"), is defined by

\forall x,y : \lang Ax | y \rang = \lang x | A^\dagger y \rang

where < | > denotes the inner product. Using the Riesz representation theorem it can be shown that:

$A^\dagger$ exists.
$A^\dagger$ is indeed a bounded linear operator.
$A^\dagger$ is determined uniquely by A. So if both B and C are Hermitian adjoints of A, then B=C.

Properties

Immediate properties:

${( A^\dagger )}^\dagger = A$
$(A + B )^\dagger = A^\dagger + B^\dagger$
$( \lambda A )^\dagger = \lambda^* A^\dagger$
$(AB)^\dagger = B^\dagger A^\dagger$

These four properties define star-algebra. If we define operator-induced norm by

\| A \| _{op} := \sup \{ \|Ax \| : \| x \| \le 1 \}

then

\| A^\dagger \| _{op} = \| A \| _{op}

Moreover,

\| A^\dagger A \| _{op} = \| A \| _{op}^2

The last formula is very useful for computing a norm of operator since

A^\dagger A

is self-adjoint (Hermitian).

Hermitian operators

An operator is called Hermitian or self-adjoint if

A = A^\dagger

which is equivalent to

\forall x,y : \lang Ax | y \rang = \lang x | A y \rang

Hermitian operators are very important because of their properties. When working in separable Hilbert space we are granted that:

$\lang x | A | x \rang$ is real for every vector x.
All of its eigenvalues are real; i.e., if $Ax = \lambda x$ than $\lambda$ is real.
If $Ax = c_1 x$ and $Ay = c_2 y$ are eigenvectors of different eigenvalues, they are orthogonal, i.e., $\lang x | y \rang = 0$ .
Every Hermitian operator has an orthonormal basis $\{ e_n \}_{n=1}^{\infty}$ of eigenvectors, such as

\forall n=1,2,3,... \ : \quad Ae_n = \lambda_n e_n

This important theorem is called orthogonal diagonalization and known as the Spectral theorem.

The following properties are the basis for Fourier expansion in Hilbert space. In Quantum mechanics, every observable is described with a corresponding Hermitian operator. The Hamiltonian (energy) of physical systems is the most important Hermitian operator, since it governs the development of the system over time.

Matrix (of finite-dimensional linear space)

When working on a finite-dimensional linear space, any linear operator (i.e. linear transformation) can be represented by a matrix. The hermitian adjoint of a complex matrix is the transpose of the matrix of the complex conjugates of the elements of the original matrix:

A^H=(A^*)^T

The concept is generalized from matrices of (real and/or complex) numbers to linear operators generally, and from vector spaces to function spaces. In particular, in separable Hilbert spaces, the conjugate and transpose operators commute so that

A^H=(A^*)^T=(A^T)^*

If an operator is sesquilinear, the hermitian adjoint of that operator is the operator itself.

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