A real, square matrix is orthogonal if (the identity matrix). Equivalently, . The columns of an orthogonal matrix are orthonormal, that is, they have 2-norm (Euclidean length) and are mutually orthogonal. The same is true of the rows.
Important examples of orthogonal matrices are rotations and reflectors. A rotation matrix has the form
For such a matrix, and for some , and the multiplication for a vector represents a rotation through an angle radians. An rotation matrix is formed by embedding the matrix into the identity matrix of order .
A Householder reflector is a matrix of the form , where is a nonzero -vector. It is orthogonal and symmetric. When applied to a vector it reflects the vector about the hyperplane orthogonal to . For , such a matrix has the form
Here is the Householder reflector corresponding to :
This is times a Hadamard matrix.
Various explicit formulas are known for orthogonal matrices. For example, the matrices with elements
are orthogonal. These and other orthogonal matrices, as well as diagonal scalings of orthogonal matrices, are constructed by the MATLAB function
Here are some properties of orthogonal matrices.
- All the eigenvalues are on the unit circle, that is, they have modulus .
- All the singular values are .
- The -norm condition number is , so orthogonal matrices are perfectly conditioned.
- Multiplication by an orthogonal matrix preserves Euclidean length: for any vector .
- The determinant of an orthogonal matrix is . A rotation has determinant while a reflection has determinant .
Orthogonal matrices can be generated from skew-symmetric ones. If is skew-symmetric () then (the matrix exponential) is orthogonal and the Cayley transform is orthogonal as long as has no eigenvalue equal to .
Unitary matrices are complex square matrices for which , where is the conjugate transpose of . They have analogous properties to orthogonal matrices.