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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