unitary

Unitary matrix

A unitary/orthogonal matrix $U$ is a complex square matrix with orthonormal column vectors.
A matrix $U\in {\mathbb{C}}^{n\times n}$ is unitary if and only if:

$$U^{†}U=U{U}^{†}=I⟹Q^{†}=Q^{-1}$$

where $U^{†}$ is the conjugate transpose (adjoint) of $U$, which equals the inverse matrix $U^{-1}$.

By definition, $Q{Q}^T=I$, any orthogonal matrix is also orthonormal.

Geometric meaning

Unitary transformations preserve geometry in vector spaces:
Inner products are preserved: $⟨x,y⟩=⟨Ux,Uy⟩\forall x,y\in {\mathbb{R}}^n$
They act as pure rotations (or reflections if the determinant is $-1$: ¹) of vectors, preserving lengths and angles between vectors.
So $U^{T}U$ just undoes the rotation.

Properties

Let $U$ be a unitary matrix. Then:

→ $\Vert Ux\Vert =\Vert x\Vert$ for any vector $x$: Unitary transformations preserve lengths.
→ $x\cdot y=Ux\cdot Uy=Ux\cdot Uy$ for any vectors $x,y$: Unitary transformations preserve angles – they just rotate.
→ The absolute value of the determinant $|\det (U)|=1$, since the basis vectors are orthonormal (on the unit circle/cube/… hence unitary) and this attribute is preserved (only the sign might change). → That’s equivalent to saying the eigenvectors of $U$ also lie on the unit circle in the complex plane.
→ The product of two unitary matrices is unitary.

Rotation matrix in ${\mathbb{R}}^2$

$$R(\theta )=\left[\begin{array}{cc}\cos (\theta )& -\sin (\theta )\\ \sin (\theta )& \cos (\theta )\end{array}\right]$$

is a unitary matrix that rotates vectors in ${\mathbb{R}}^2$ by an angle $\theta$ counterclockwise.
Clockwise rotation is achieved by using $-\theta$, or swapping the signs of the off-diagonal elements.
Intuitively this works because the first column is where the basis vector $(1,0)$ ends up after rotation by $\theta$, and the second column is where the basis vector $(0,1)$ ends up after rotation by $\theta$.

EXAMPLE

$$\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\cdot \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}-y\\ x\end{array}\right]$$

is a matrix that rotates vectors in ${\mathbb{R}}^2$ by $90^{\circ }$ counterclockwise.

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Common unitary matrices

permutation matrix
rotation matrix in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$
The fourier matrix used in the discrete fourier transform

Applications

Unitary matrices are fundamental in:

signal processing (preserving signal energy)
PCA and SVD (rotations in high dimensions)
error-correcting codes (preserving hamming distance)

References

Visual Kernel
SVD - Data-Driven Science and engineering - Steve Brunton

Footnotes

1. A composition between a pure rotation and a reflection is called improper rotation. So strictly speaking, orthogonal matrices always produce improper rotations. But for many cases this does not matter, e.g. in eigendecomposition, where $Q$ contains the normalized eigenvectors of $S$, then a $-v$ is just as much a normalized eigenvector as $v\in Q$, so $\pm v$ can be used interchangeably, and you can even pick the signs to ensure that $Q$ produces a pure rotation. ↩