Adjoints of Linear Transformations

Definition

Suppose $T \in \mathcal{L} (V, W)$. In other words, $T$ is a linear map from a vector space $V$ to a vector space $W$ where $V$ and $W$ are over the same field of scalars.

An adjoint of $T$ is a function $S: W \to V$ satisfying

\[ \ip{Tv, w} = \ip{v, Sw} \quad \text{for every } v \in V \text{ and every } w \in W \]

It is not hard to see that there always exists an adjoint, and that this adjoint is unique. Thus, we speak of the adjoint of $T$.

Notation

We denote the adjoint of $T$ by $T^*$. This notation is meant to remind of complex conjugation, for reasons which will become apparent shortly.

Examples

Space to the plane. Define $T: \R ^3 \to \R ^2$ by

\[ T(x_1, x_2, x_3) = (x_2 + 3x_3, 2x_1) \]

We claim that the adjoint of $T$ is $T^*: \R ^2 \to \R ^3$ defined by

\[ T^*(y_1, y_2) = (2y_2, y_1, 3y_1) \]

Properties

Suppose $T \in \mathcal{L} (V, W)$. The adjoint of $T$ is linear.

Suppose $V$ and $W$ are finite dimensional inner product spaces over a field $\F $, which is $\R $ or $\C $. Suppose $S, T \in \mathcal{L} (V, W)$. Then

$(S + T)^* = S^* + T^*$
$(\lambda T)^* = \Cconj{\lambda }T^*$ for all $\lambda \in \F $
$ (T^*)^* = T$
$ I^* = I$

Suppose $V$, $W$, $U$ are finite dimensional inner product spaces over $\R $ or $\C $. For all $T \in \mathcal{L} (V, W)$ and $S \in \mathcal{L} (W, U)$,

\[ (ST)^* = T^*S^* \]