# Contents

Suppose $ℋ$ is a Hilbert space with an inner product $⟨\cdot ,\cdot ⟩$. Consider a continuous linear operator $A:ℋ\to ℋ$. One can show that there exists a unique continuous linear operator ${A}^{*}:ℋ\to ℋ$ with the following property:

$⟨Ax,y⟩=⟨x,{A}^{*}y⟩$ for all $x,y\in ℋ$.

This is a generalization of the concept of an adjoint matrix (also known as a conjugate transpose, Hermitian conjugate, or Hermitian adjoint). The adjoint of an $m×n$ matrix $A$ with complex entries is the $n×m$ matrix whose entries are defined by

$\left({A}^{*}{\right)}_{\mathrm{ij}}=\overline{{A}_{\mathrm{ji}}}$.

As such,

${A}^{*}=\left(\overline{A}{\right)}^{T}=\overline{{A}^{T}}$

where ${A}^{T}$ is the transpose of $A$ and $\overline{A}$ is the matrix with complex conjugate entries of $A$.

## Hermitian matrices

A matrix, $A$, is said to be Hermitian if

${A}^{*}=A$

where ${A}^{*}$ is the Hermitian adjoint of $A$.

## Notation

The notation used here for the adjoint, ${A}^{*}$, is commonly used in linear algebraic circles (as is ${A}^{H}$). In quantum mechanics, ${A}^{†}$ is exclusively used for the adjoint while ${A}^{*}$ is interpreted as the same thing as $\overline{A}$.

Revised on September 5, 2010 03:59:19 by Toby Bartels (173.190.157.63)