What Is an Orthogonal Matrix?

A real, square matrix $Q$ is orthogonal if $Q^TQ = QQ^T = I$ (the identity matrix). Equivalently, $Q^{-1} = Q^T$ . The columns of an orthogonal matrix are orthonormal, that is, they have 2-norm (Euclidean length) $1$ and are mutually orthogonal. The same is true of the rows.

Important examples of orthogonal matrices are rotations and reflectors. A $2\times 2$ rotation matrix has the form

$\begin{bmatrix} c & s \\ -s& c \\ \end{bmatrix}, \quad c^2 + s^2 = 1.$

For such a matrix, $c = \cos \theta$ and $s = \sin \theta$ for some $\theta$ , and the multiplication $y = Qx$ for a $2\times 1$ vector $x$ represents a rotation through an angle $\theta$ radians. An $n\times n$ rotation matrix is formed by embedding the $2\times 2$ matrix into the identity matrix of order $n$ .

A Householder reflector is a matrix of the form $H = I - 2uu^T/(u^Tu)$ , where $u$ is a nonzero $n$ -vector. It is orthogonal and symmetric. When applied to a vector it reflects the vector about the hyperplane orthogonal to $v$ . For $n = 2$ , such a matrix has the form

$\begin{bmatrix} c & s \\ s& -c \\ \end{bmatrix}, \quad c^2 + s^2 = 1.$

Here is the $4\times 4$ Householder reflector corresponding to $v = [1,1,1,1]^T/2$ :

$\frac{1}{2} \left[\begin{array}{@{\mskip2mu}rrrr@{\mskip2mu}} 1 & -1 & -1 & -1\\ -1 & 1 & -1 & -1\\ -1 & -1 & 1 & -1\\ -1 & -1 & -1 & 1\\ \end{array}\right].$

This is $1/2$ times a Hadamard matrix.

Various explicit formulas are known for orthogonal matrices. For example, the $n\times n$ matrices with $(i,j)$ elements

$q_{ij} = \displaystyle\frac{2}{\sqrt{2n+1}} \sin \left(\displaystyle\frac{2ij\pi}{2n+1}\right)$

and

$q_{ij} = \sqrt{\displaystyle\frac{2}{n}}\cos \left(\displaystyle\frac{(i-1/2)(j-1/2)\pi}{n} \right)$

are orthogonal. These and other orthogonal matrices, as well as diagonal scalings of orthogonal matrices, are constructed by the MATLAB function gallery('orthog',...).

Here are some properties of orthogonal matrices.

All the eigenvalues are on the unit circle, that is, they have modulus $1$ .
All the singular values are $1$ .
The $2$ -norm condition number is $1$ , so orthogonal matrices are perfectly conditioned.
Multiplication by an orthogonal matrix preserves Euclidean length: $\|Qx\|_2 = \|x\|_2$ for any vector $x$ .
The determinant of an orthogonal matrix is $\pm 1$ . A rotation has determinant $1$ while a reflection has determinant $-1$ .

Orthogonal matrices can be generated from skew-symmetric ones. If $S$ is skew-symmetric ( $S = -S^T$ ) then $\exp(S)$ (the matrix exponential) is orthogonal and the Cayley transform $(I-S)(I+S)^{-1}$ is orthogonal as long as $S$ has no eigenvalue equal to $-1$ .

Unitary matrices are complex square matrices $Q$ for which $Q^*Q = QQ^* = I$ , where $Q^*$ is the conjugate transpose of $Q$ . They have analogous properties to orthogonal matrices.

Related Blog Posts

What Is a Hadamard Matrix? (2020)—forthcoming
What Is a Random Orthogonal Matrix? (2020)—forthcoming

This article is part of the “What Is” series, available from https://nhigham.com/category/what-is and in PDF form from the GitHub repository https://github.com/higham/what-is.

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