What is "orthogonalization"

n. 1 (context mathematics English) The process of converting a set of functions or vectors into orthogonal ones 2 (context colloquial English) the process of splitting a problem or system into its distinct components

In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors {v, ... , v} in an inner product space (most commonly the Euclidean space R), orthogonalization results in a set of orthogonal vectors {u, ... , u} that generate the same subspace as the vectors v, ... , v. Every vector in the new set is orthogonal to every other vector in the new set; and the new set and the old set have the same linear span.

In addition, if we want the resulting vectors to all be unit vectors, then the procedure is called orthonormalization.

Orthogonalization is also possible with respect to any symmetric bilinear form (not necessarily an inner product, not necessarily over real numbers), but standard algorithms may encounter division by zero in this more general setting.