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Orthogonality is the generalization of the geometric notion of perpendicularity. This term is used to describe sets and matrices with special properties that its elements form an orthonormal basis. The basic idea is that the axes are perpendicular to each other and have unit length.

Orthogonal Sets

Orthogonal Matrices

Orthogonal transformations and correponding matrices are interesting because it is easy to compute their inverse and they arise frequently in practice. Translation, rotation, and reflection are the only orthogonal transformations. All orthogonal transformations are affine and invertible. Lengths, angles, areas, and volumes are all preserved. However, e aware that reflections inverse directions of agles.
A square matrix Q ∈ ℝn×n is called orthogonal or orthonormal matrix, if its rows are orthonormal sets of vectors: mutual perpendicular and of length 1. A linear transformation is said to be orthogonal if its matrix is.
Example 1: Obviously, the identity matrix I is an orthogonal matrix.

Rotation matrixc in ℝ² is another example of an orthogonal matrix: \[ \begin{bmatrix} \cos\theta & - \sin]theta \\ \sin\theta & \cos\theta \end{bmatrix} . \]

Reflections are the elements of O(n) whose canonical form is \[ \begin{bmatrix} -1 & {\bf 0} \\ {\bf 0} & {\bf I} \end{bmatrix} , \] where I = In-1 is the (n–1)×(n–1) identity matrix, and the zeros denote row or column zero matrices. In other words, a reflection is a transformation that transforms the space in its mirror image with respect to a hyperplane.

End of Example 1
The set of n × n orthogonal matrices, under multiplication, forms the group O(n), known as the orthogonal group. The subgroup SO(n) consisting of orthogonal matrices with determinant +1 is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation.

Theorem 1: For n × n matrix Q ∈ ℝn×n and for linear transformation T given by \( \displaystyle T({\bf x}) = {\bf x}\,{\bf Q} , \) the following statements are equivalent.

  1. Q is orthogonal, i.e., the rows of Q are orthonormal.
  2. QT is orthogonal, i.e., the columns of Q are orthonormal.
  3. Q−1 = QT.
  4. Q QT = I.
  5. QTQ = I.
  6. T preserves dot products, namely, T(x) • T(y) = x • y.
  7. T preserves lengths, that is, |T(x)| = |x| for all x.
  8. T preserves distances, namely, ∥T(x) − T(y)∥ = ∥x − y∥.

Corollary 1: The product of two oethogonal matrices is an orthogonal matrix.

A rotation matrix is an orthogonal matrix of determinant +1; so it is element of SO(n).