28 - Equivalent Statements of Adjoint Map

Equivalent Statements of Adjoint Map

Let $T$ be a linear map on $V$ then

We have the following equivalent statements

1. $T$ adjoint is the inverse

$$T^{\ast }=T^{-1}$$

2. $T$ preserves inner products so

$$⟨v,w⟩=⟨Tv,Tw⟩\text{ for all }v,w\in V$$

3. $T$ preserves lengths so

$$||v||=||Tv||\text{ for all }v\in V$$

Proof

$$\begin{array}{rlrlrl}(1)⟹(2)& ⟨v,w⟩& & =⟨\mathrm{Id}v,w⟩& & \\ & & & =⟨T^{\ast }Tv,w⟩& & \\ & & & =⟨Tv,Tw⟩,& & \text{ for all }v,w\in V\\ & & & & & \\ (2)⟹(3)& ||v|{|}^2& & =⟨v,v⟩& & \\ & & & =⟨Tv,Tv⟩& & \\ & & & =||Tv|{|}^2,& & \text{ for all }v\in V\\ & & & & & \\ (2)⟹(1)& ⟨v,w⟩& & =⟨Tv,Tw⟩& & \\ & & & =⟨T^{\ast }Tv,w⟩,& & \text{ for all }v,w\in V\\ & & & ⟹T^{\ast }Tv=v& & \text{ by non-degeneracy of }⟨,⟩,\text{ for all }v\in V\\ & & & ⟹T^{\ast }T=\mathrm{Id}& & \end{array}$$

For $(3)⟹(2)$ refer to length determines inner product