Orthogonal Matrix

An orthogonal matrix is an real matrix satisfying

(1)

where is the transpose of and is the identity matrix. In particular, an orthogonal matrix is always invertible, and

(2)

In component form,

(3)

This relation makes orthogonal matrices particularly easy to compute with, since the transpose operation is much simpler than computing a matrix inverse.

For example,

			(4)
			(5)

are orthogonal matrices.

A matrix can be tested to see if it is orthogonal in the Wolfram Language using OrthogonalMatrixQ[m].

The rows of an orthogonal matrix are an orthonormal basis. That is, each row has length one, and the rows are mutually perpendicular. Similarly, the columns are also an orthonormal basis. In fact, given any orthonormal basis, the matrix whose rows are that basis is an orthogonal matrix. It is automatically the case that the columns are another orthonormal basis.

The orthogonal matrices are precisely those matrices which preserve the inner product

(6)

Also, the determinant of is either 1 or . As a subset of , the orthogonal matrices are not connected since the determinant is a continuous function. Instead, there are two components corresponding to whether the determinant is 1 or . The orthogonal matrices with are rotations, and such a matrix is called a special orthogonal matrix.

The matrix product of two orthogonal matrices is another orthogonal matrix. In addition, the matrix inverse of an orthogonal matrix is an orthogonal matrix, as is the identity matrix. Hence the set of orthogonal matrices forms a group, called the orthogonal group .