Linear and multilinear algebra; matrix theory Definition

Adjoint of a linear map

Builds on Inner product space Linear map

Definition

Let $V$ and $W$ be finite-dimensional inner product spaces over $\R$ or $\C$ , and let $T \colon V \to W$ be a linear map. The adjoint of $T$ is the unique linear map $T^* \colon W \to V$ such that $\langle T(v), w \rangle = \langle v, T^*(w) \rangle \qquad \text{for all } v \in V \text{ and } w \in W.$

Remarks

The existence and uniqueness follow from the Riesz representation theorem. It satisfies $(S + T)^* = S^* + T^*$ , $(aT)^* = \bar a T^*$ , $(ST)^* = T^* S^*$ and $T^{**} = T$ . In orthonormal bases, the matrix of $T^*$ is the conjugate transpose of the matrix of $T$ . It satisfies $\ker T^* = (\im T)^\perp$ .

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Vector space definition
Inner product definition
Linear map definition
Inner product space definition
Adjoint of a linear map definition

Tag	`0ST8QF9T3T`
URI	`m2://0ST8QF9T3T`
Kind	definition
Writing	W1 stated
MSC	15A04, 47A05
Defines	Adjoint of a linear map
Uses	Inner product space, Linear map

Adjoint of a linear map

Definition

Remarks

Reading path

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