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hermitian matrix

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Hermitian adjoints

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

Ax,y=x,A *y\langle A x,y\rangle=\langle x,A^{*}y\rangle for all x,yx,y \in \mathcal{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×nm \times n matrix AA with complex entries is the n×mn \times m matrix whose entries are defined by

(A *) ij=A ji¯(A^{*})_{ij}=\bar{A_{ji}}.

As such,

A *=(A¯) T=A T¯A^{*}=(\bar{A})^{T}=\bar{A^{T}}

where A TA^{T} is the transpose of AA and A¯\bar{A} is the matrix with complex conjugate entries of AA.

Hermitian matrices

A matrix, AA, is said to be Hermitian if

A *=AA^{*}=A

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

Notation

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

Revised on October 30, 2013 22:03:13 by Urs Schreiber (82.169.114.243)