Hermitian adjoint

Conjugate transpose of an operator in infinite dimensions / From Wikipedia, the free encyclopedia

Dear Wikiwand AI, let's keep it short by simply answering these key questions:

Can you list the top facts and stats about Hermitian adjoint?

Summarize this article for a 10 years old


In mathematics, specifically in operator theory, each linear operator on an inner product space defines a Hermitian adjoint (or adjoint) operator on that space according to the rule

where is the inner product on the vector space.

The adjoint may also be called the Hermitian conjugate or simply the Hermitian[1] after Charles Hermite. It is often denoted by A in fields like physics, especially when used in conjunction with bra–ket notation in quantum mechanics. In finite dimensions where operators are represented by matrices, the Hermitian adjoint is given by the conjugate transpose (also known as the Hermitian transpose).

The above definition of an adjoint operator extends verbatim to bounded linear operators on Hilbert spaces . The definition has been further extended to include unbounded densely defined operators whose domain is topologically dense in - but not necessarily equal to -