Orthogonal and Unitary Matrices

What Makes a Matrix Orthogonal or Unitary

A real square matrix $Q\in\mathbb{R}^{n\times n}$ is orthogonal when

Q^TQ=QQ^T=I

A complex square matrix $U\in\mathbb{C}^{n\times n}$ is unitary when

U^HU=UU^H=I

Here $U^H=\overline{U}^{T}$ is the conjugate transpose. A real orthogonal matrix is a $unitary matrix with real entries$ .

The defining equations are equivalent to saying that the columns form an orthonormal basis. The rows are orthonormal as well.

What Orthogonal and Unitary Transformations Preserve

For real vectors $x,y$ and an orthogonal matrix $Q$ ,

\begin{aligned} \langle Qx,Qy\rangle &=(Qx)^T(Qy) \\ &=x^TQ^TQy \\ &=x^Ty \end{aligned}

The same statements hold over $\mathbb{C}$ with conjugate inner products and $U^H$ . Consequently, orthogonal and unitary transformations $preserve lengths, distances, angles, and orthogonality$ .

All their singular values equal $1$ , so

\|Q\|_2=1,\qquad\kappa_2(Q)=1

Multiplication by such a matrix does not amplify relative errors in the Euclidean norm. This is why $\mathrm{QR}$ factorizations rely on Householder reflectors and Givens rotations.

Determinants and Orientation

For a unitary matrix,

\begin{aligned} 1 &=\det(U^HU) \\ &=\overline{\det U}\det U \\ &=|\det U|^2 \end{aligned}

For a real orthogonal matrix, the determinant is real, so

\det Q\in\{-1,1\}

The case $\det Q=1$ $preserves orientation$ . The case $\det Q=-1$ reverses orientation and contains a reflection component.

Eigenvalues and Spectral Structure

If $Uv=\lambda v$ with $v\ne0$ , then

\|v\|_2=\|Uv\|_2=\|\lambda v\|_2=|\lambda|\|v\|_2

Every eigenvalue therefore lies on the $complex unit circle$ . Nonreal eigenvalues of a real orthogonal matrix occur in conjugate pairs. A unitary matrix is normal, so the complex spectral theorem gives

U=V\operatorname{diag}(e^{i\theta_1},\ldots,e^{i\theta_n})V^H

In this decomposition, $V$ is unitary. Over the real numbers, an orthogonal matrix can instead be reduced to blocks consisting of $1$ , $-1$ , and planar rotations.

Rotation Example

A counterclockwise planar rotation by $\alpha$ is

R(\alpha)=\begin{pmatrix}\cos\alpha&-\sin\alpha\\\sin\alpha&\cos\alpha\end{pmatrix}

Its columns are orthonormal, and direct multiplication gives

R(\alpha)^TR(\alpha)=I

The characteristic polynomial is

\chi_R(t)=t^2-2t\cos\alpha+1

Unless $\alpha$ is a multiple of $\pi$ , the rotation has $no real eigenvector$ . That is expected: no nonzero real direction remains fixed or merely changes sign under a genuine planar rotation.

Reflection Example

For a unit vector $u\in\mathbb{R}^n$ , the Householder matrix

H=I-2uu^T

The matrix $H$ reflects a vector across the $hyperplane perpendicular$ to $u$ . Its orthogonality can be checked from

H^T=H

The two identities above give $H^TH=I$ . Its eigenvalue in the direction $u$ is $-1$ , while every vector perpendicular to $u$ has eigenvalue $1$ . Therefore $\det H=-1$ .

Equivalent Checks and Consequences

Statement	Orthogonal case	Unitary case
Inverse	$Q^{-1}=Q^T$	$U^{-1}=U^H$
Columns	real orthonormal basis	complex orthonormal basis
Determinant	$\pm1$	modulus $1$
Eigenvalues	unit circle, conjugate pairs when nonreal	unit circle
Singular values	all $1$	all $1$
Two-norm condition number	$1$	$1$

$Products and inverses$ remain in the same class:

(Q_1Q_2)^T(Q_1Q_2)=I

Because orthogonal and unitary matrices preserve norms, the orthogonal transformations used in $\mathrm{QR}$ and least-squares calculations do not change vector lengths. Fourier transforms, quantum mechanics, and geometry processing use the same property to preserve inner products and norms.