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Orthogonal

Redirected from Orthogonality In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. It means at right angles. It comes from the Greek "ortho", meaning "right" and "gonia", meaning "angle". Two streets that cross each other at a right angle are orthogonal to each other. Two vectors in an inner product space are orthogonal if their inner product is zero. The word normal is sometimes also used for this concept by mathematicians, although that word is rather overburdened.

For example, in a 2- or 3-dimensional Euclidean space, two vectors are orthogonal if their dot product is zero, i.e., they make an angle of 90 degree. Hence orthogonality is a generalization of the concept of perpendicular.

Several vectors are called pairwise orthogonal if any two of them are orthogonal, and a set of such vectors is called an orthogonal set. They are said to be orthonormal if they are all unit vectors. Non-zero pairwise orthogonal vectors are always linearly independent.

Functions may also be orthogonal with respect to some nonnegative weight function $w(x)$ in the sense that their inner products $<f_i,f_j>$ are

$<f_i, f_j> = \int_a^b w(x)f_i(x)f_j(x) dx = 0\quad\forall i\not=j$