Isometries

Aug-Nov 2020

Recap

• Vector space $V$ over a scalar field $F= \mathbb{R}$ or $\mathbb{C}$
• $m\times n$ matrix A represents a linear map $T:F^n\to F^m$
• dim null $T+$ dim range $T=$ dim $V$
• Solution to $Ax=b$ (if it exists): $u+$ null$(A)$
• Four fundamental subspaces of a matrix
• Column space, row space, null space, left null space
• Eigenvalue $\lambda$ and Eigenvector $v$: $Tv=\lambda v$
• There is a basis w.r.t. which a linear map is upper-triangular
• If there is a basis of eigenvectors, linear map is diagonal w.r.t. it
• Inner products, norms, orthogonality and orthonormal basis
• There is an orthonormal basis w.r.t. which a linear map is upper-triangular
• Orthogonal projection: distance from a subspace
• Adjoint of a linear map: $\langle Tv,w\rangle=\langle v,T^*w\rangle$
• null $T=$ $($range $T^*)^{\perp}$
• Self-adjoint: $T=T^*$, Normal: $TT^*=T^*T$
• Eigenvectors corresponding to different eigenvalues are orthogonal
• Complex/real spectral theorem: $T$ is normal/self-adjoint $\leftrightarrow$ orthonormal basis of eigenvectors
• Positive operators: self-adjoint with non-negative eigenvalues

Rotation and reflection in two dimensions

Rotation by angle $\theta$

$R=\begin{bmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}$

Eigenvalues: $e^{i\theta}$, $e^{-i\theta}$

No real eigenspace (no axis of rotation)

Reflection about $x$-axis

$S=\begin{bmatrix} 1&0\\ 0&-1\end{bmatrix}$

Eigenvalues: 1 and -1

Eigenspaces: $x$-axis is fixed, $y$-axis is flipped

Norms before and after transform
$\lVert Rx\rVert=\lVert x\rVert$

$\lVert Sx\rVert=\lVert x\rVert$

Definition of isometries

$V$: inner product space over $F=\mathbb{R}$ or $\mathbb{C}$

An operator $T:V\to V$ is called an isometry if $\lVert Tv\rVert=\lVert v\rVert$ for all $v$

Examples

1. $A=\begin{bmatrix} \lambda_1&&0\\ &\ddots&\\ 0&&\lambda_n \end{bmatrix}$ with $\lvert\lambda_i\rvert=1$

2. $A=\begin{bmatrix} 1&0&0\\ 0&\cos\theta&-\sin\theta\\ 0&\sin\theta&cos\theta \end{bmatrix}$

Also known as

Real spaces: orthogonal operators

Complex spaces: unitary operators

When is a normal operator an isometry?

$F=\mathbb{C}$, $T$: normal operator

$T=\lambda_1 e_1\overline{e^T_1}+\cdots+\lambda_n e_n\overline{e^T_n}$

$\{e_1,\ldots,e_n\}$: orthonormal

Let $x=x_1e_1+\cdots+x_ne_n$, $\lVert x\rVert^2=\lvert x_1\rvert^2+\cdots+\lvert x_n\rvert^2$

$\lVert Tx\rVert^2=\lvert\lambda_1\rvert^2\lvert x_1\rvert^2+\cdots+\lvert\lambda_n\rvert^2\lvert x_n\rvert^2$

A normal operator is an isometry if every eigenvalue has absolute value 1.

Interesing result: There are no other isometries

Characterising isometries

$V$: inner product space over $F=\mathbb{R}$ or $\mathbb{C}$

Suppose $S:V\to V$ is an operator. The following are equivalent:

1. $S$ is an isometry
2. $\langle Su,Sv\rangle=\langle u,v\rangle$
3. If $\{e_1,\ldots,e_n\}$ is orthonormal, so is $\{Se_1,\ldots,Se_n\}$
4. There exists one orthonormal basis $\{e_1,\ldots,e_n\}$ s.t. $\{Se_1,\ldots,Se_n\}$ is orthonormal
5. $S^*S=SS^*=I$ ($S$ is invertible and $S^{-1}=S^*$)
6. $S^*$ is an isometry

Proof of (1) implies (2)

Real: $\langle u,v\rangle=\dfrac{\lVert u+v\rVert^2-\lVert u-v\rVert^2}{4}$

Complex: $\langle u,v\rangle=\dfrac{\lVert u+v\rVert^2-\lVert u-v\rVert^2+i\lVert u+iv\rVert^2-i\lVert u-iv\rVert^2}{4}$

Proof (continued)

Proof of (2) implies (3), (4)

$\langle Se_i,Se_j\rangle=\langle e_i,e_j\rangle$

Proof of (4) implies (5)

$\langle S^*Se_i,e_j\rangle=\langle Se_i,Se_j\rangle=\langle e_i,e_j\rangle$ for all $i,j$

For $u=u_1e_1+\cdots+u_ne_n$, $v=v_1e_1+\cdots+v_ne_n$,

$\begin{gather} \langle S^*Su,v\rangle\\[5pt] =u_1\overline{v_1}\langle S^*Se_1,e_1\rangle+\cdots+u_i\overline{v_j}\langle S^*Se_i,e_j\rangle+\cdots+u_n\overline{v_n}\langle S^*Se_n,e_n\rangle\\[5pt] =u_1\overline{v_1}\langle e_1,e_1\rangle+\cdots+u_i\overline{v_j}\langle e_i,e_j\rangle+\cdots+u_n\overline{v_n}\langle e_n,e_n\rangle\\[5pt] =\langle u,v\rangle\end{gather}$

So, $S^*S=I$, which implies $SS^*=I$, $S$ is invertible and $S^{-1}=S^*$

Completing the proof and corollaries

Proof of (5) implies (6), (1)

$\lVert S^*v\rVert^2=\langle S^*v,S^*v\rangle=\langle SS^*v,v\rangle=\langle v,v\rangle=\lVert v\rVert^2$

$\lVert Sv\rVert^2=\langle Sv,Sv\rangle=\langle S^*Sv,v\rangle=\langle v,v\rangle=\lVert v\rVert^2$

Corollary: Every isometry is normal

Proof: $SS^*=S^*S=I$

Corollary: A matrix represents an isometry if and only if (1) rows and columns have unit norm, (2) any two rows or columns are orthogonal.

Isometries in complex spaces

$V$: complex inner product space, $S:V\to V$

$S$ is an isometry if and only if there exists an orthonormal basis of eigenvectors of $S$ with every eigenvalue having an absolute value 1.

Proof

Forward result

Normal with $\lvert\lambda_i\rvert=1$ implies isometry (example)

Converse

If $S$: isometry, then $S$ is normal and has an orthonormal eigenvector basis $\{e_1,\ldots,e_n\}$

Let $Se_i=\lambda_i e_i$

Taking norms,

$\lvert\lambda\rvert=\lVert\lambda_i e_i\rVert=\lVert Se_i\rVert=\lVert e_i\rVert = 1$