Inner product/Adjoint endomorphism/Definition

Adjoint endomorphism

Let ${}V$ be a ${}{\mathbb {K} }$-vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$, and let

$\varphi \colon V\longrightarrow V$

denate an endomorphism. An endomorphism

$\psi \colon V\longrightarrow V$

is called adjoint to ${}\varphi$, if

${}\left\langle \varphi (v),w\right\rangle =\left\langle v,\psi (w)\right\rangle \,$

holds for all ${}v,w\in V$.