# 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