19 - Orthogonal and Unitary

Orthogonal and Unitary

Let $V$ be a finite dimensional inner product space
Let $T:V\to V$ be a linear transformation then

If $T^{\ast }=T^{-1}$ then $T$ is called

$$\begin{array}{r}\text{orthogonal}\text{ when }\mathbb{K}=\mathbb{R}\\ \text{unitary}\text{ when }\mathbb{K}=\mathbb{C}\end{array}$$

Matrix Equivalence of Orthogonal and Unitary

Let $\mathcal{B}$ be an orthonormal basis for $V$
Let $T$ be an orthogonal / unitary transformation of $V$ then

$${}_{\mathcal{B}}[T{]}_{\mathcal{B}}\text{ is a orthogonal / unitary matrix}$$

using matrix representation of adjoint

Eigenvalues of Orthogonal / Unitary Linear Transformation lemma

Let $\lambda$ be an eigenvalue of an orthogonal / unitary linear transformation $T:V\to V$ then

$$|\lambda |=1$$

Proof

Let $v\ne 0$ be a $\lambda$-eigenvector then

$$⟨v,v⟩=⟨Tv,Tv⟩$$

by Equivalent Statements of Adjoint Map then

$$⟨Tv,Tv⟩=⟨\lambda v,\lambda v⟩=\overline{\lambda }\lambda ⟨v,v⟩$$

Hence $1=\overline{\lambda }\lambda$ so $|\lambda |=1$

Determinant of Orthogonal / Unitary Matrix corollary

Let $A$ be an orthogonal / unitary $n\times n$ matrix then

$$|\det A|=1$$

Proof

As we are working over $\mathbb{C}$ and $\det A$ is the product of all eigenvalues (with repetitions)
Hence

$$\det A=|{\lambda }_1{\lambda }_2\cdots {\lambda }_n|=|{\lambda }_1||{\lambda }_2|\cdots |{\lambda }_n|=1$$

Orthogonal Complements of Invariant Subspaces for Unitary Operators lemma

Assume that $V$ is finite dimensional and $T:V\to V$ with $T^{\ast }T=\mathrm{Id}$ then

$$U\text{ is }T-\text{invariant}⟹U^{\perp }\text{ is }T-\text{invariant}$$

Proof

For $w\in U^{\perp }$ then for all $u\in U$ then

$$⟨u,Tw⟩=⟨T^{\ast }u,w⟩=⟨T^{-1}u,w⟩$$

As $U$ is invariant under $T$ then it must be invariant under $T^{-1}=T^{\ast }$
This can be seen by writing

$$m_T(x)=x^m+a_{m-1}{x}^{m-1}+\cdots +a_{1}x+a_0$$

into

$$T(T^{m-1}+a_{m-2}{T}^{m-2}+\cdots +a_1)=-a_{0}I$$

where $T^{-1}$ can be written as a polynomial in terms of $T$ so

So $T^{\ast }u\in U$ and $⟨T^{\ast }u,w⟩=0$ for all $u\in U$ hence
Hence

$$⟨u,Tw⟩=0\text{ for all }u\in U⟹Tw\in U^{\perp }$$

Unitary Diagonalisation corollary

Let $A\in U(n)$ then

$$\exists P\in U(n)\text{ such that }{P}^{-1}AP\text{ is diagonal}$$