Adjoint Operators and Matrices

Def: Suppose $T \in L (V, W)$ . The adjoint of $T$ is the function $T^{*} : W \to V$ such that

⟨ T v, w ⟩ = ⟨ v, T^{*} w ⟩

for every $v \in V$ and every $w \in W$ .

If $T \in L (V, W)$ , then $T^{*} \in L (W, V)$ .

Properties
$(S + T)^{*} = S^{*} + T^{*}$ , for all $S, T \in L (V, W)$
$(λ T)^{*} = \overset{―}{λ} T^{*}$ , for all $λ \in F$ and $T \in L (V, W)$
$(T^{*})^{*} = T$ for all $T \in L (V, W)$
$I^{*} = I$ , where $I$ is the identity operator on $V$
$(S T)^{*} = T^{*} S^{*}$ for all $T \in L (V, W)$ and $S \in L (W, U)$

Th: For arbitrary vector spaces $V$ and $W$ , and $T \in L (V, W)$ then

$N (T^{*}) = R (T)^{⊥}$
$N (T) = R (T^{*})^{⊥}$
$(*) {cl}_{V} (R (T^{*})) = N (T)^{⊥}$
$(*) {cl}_{W} (R (T)) = N (T^{*})^{⊥}$
$R (T^{*}) \subseteq N (T)^{⊥}$
$R (T) \subseteq N (T^{*})^{⊥}$

Some special cases are:

$R (T^{*}) = N (T)^{⊥}$ if $V$ is finite dimensional
$R (T) = N (T^{*})^{⊥}$ if $W$ is finite dimensional

Th: Let $V$ be a finite dimensional inner product space, and $β$ be an orthonormal basis for $V$ . If $T \in L (V)$ , then

[T^{*}]_{β} = [T]_{β}^{*}

Where $[T]_{β}^{*}$ is the conjugate transpose of $[T]_{β}$ .
Cor: Let $A \in M_{n} (C)$ , then $L_{A^{*}} = (L_{A})^{*}$

Cor: If $A \in M_{m \times n} (F)$ , $x \in F^{n}$ , and $y \in F^{m}$ . Then

⟨ A x, y ⟩_{m} = ⟨ x, A^{*} y ⟩_{n}

Th: Let $A \in M_{m \times n} (F)$ . Then $rank (A^{*} A) = rank (A)$

Cor: If $A \in M_{m \times n} (F)$ such that $rank (A) = n$ , then $A^{*} A$ is invertible.

Th: Let $V$ be finite dimensional and $T \in L (V)$ . If $T$ has eigenvector, then so does $T^{*}$ . Additionally, if $λ$ is an eigenvalue of $T$ , then $\overset{―}{λ}$ is an eigenvalue of $T^{*}$ .

Th: Suppose $V$ is a complex inner product space and $T \in L (V)$ . If for all $v \in V$

⟨ T v, v ⟩ = 0

Then $T = 0$ .