Orthogonal matrix

Thus finite-dimensional linear isometries—rotations, reflections, and their combinations—produce orthogonal matrices. The converse is also true: orthogonal matrices imply orthogonal transformations. However, linear algebra includes orthogonal transformations between spaces which may be neither finite-dimensional nor of the same dimension, and these have no orthogonal matrix equivalent.

Below are a few examples of small orthogonal matrices and possible interpretations.

The simplest orthogonal matrices are the 1 × 1 matrices [1] and [−1], which we can interpret as the identity and a reflection of the real line across the origin.

Regardless of the dimension, it is always possible to classify orthogonal matrices as purely rotational or not, but for 3 × 3 matrices and larger the non-rotational matrices can be more complicated than reflections. For example,

However, we have elementary building blocks for permutations, reflections, and rotations that apply in general.

The determinant of any orthogonal matrix is +1 or −1. This follows from basic facts about determinants, as follows:

The converse is not true; having a determinant of ±1 is no guarantee of orthogonality, even with orthogonal columns, as shown by the following counterexample.

With permutation matrices the determinant matches the signature, being +1 or −1 as the parity of the permutation is even or odd, for the determinant is an alternating function of the rows.

For example, consider a non-orthogonal matrix for which the simple averaging algorithm takes seven steps

Gram-Schmidt yields an inferior solution, shown by a Frobenius distance of 8.28659 instead of the minimum 8.12404.

This may be combined with the Babylonian method for extracting the square root of a matrix to give a recurrence which converges to an orthogonal matrix quadratically:

Using a first-order approximation of the inverse and the same initialization results in the modified iteration:

The Pin and Spin groups are found within Clifford algebras, which themselves can be built from orthogonal matrices.

There is no standard terminology for these matrices. They are variously called "semi-orthogonal matrices", "orthonormal matrices", "orthogonal matrices", and sometimes simply "matrices with orthonormal rows/columns".