Orthogonal

Orthogonality is a concept defined in mathematics for spaces in which a dot product or inner product is defined. Two entities in that space are orthogonal if their dot product is zero.

In common speech (in an Euclidean space) two lines are orthogonal if they form a right angle, i.e. if the angle between them is 90 degrees. In mathematical terms this means that the dot product of two vectors aligned to the lines is zero. Similarly, a line is said to be orthogonal to a plane if the two form a right angle, and two planes are said to be orthogonal if they intersect in a right angle. The term perpendicular is also used to describe this relationship.

A square matrix A with real entries is called orthogonal if its inverse is equal to its transpose. This is equivalent to stating that its rows form an orthonormal basis of Rⁿ with the ordinary Euclidean dot product. All eigenvalues of such a matrix A have absolute value 1. Furthermore, for every vector x in Rⁿ, the Euclidean norm of Ax is the same as that of x. A matrix can be made orthogonal using the Gram-Schmidt Process.