ortogonaliset

Ortogonaliset is a term used in linear algebra to describe the process of converting a collection of vectors into an orthogonal, and often orthonormal, set with respect to a given inner product. The concept underpins many computational methods because orthogonality simplifies projection, decomposition, and numerical stability.

In formal terms, in a real or complex inner product space, given a linearly independent set {v1, ..., vk}, orthogonalization produces {u1, ..., uk} with ui in span(v1, ..., vi) and inner products <ui, uj> = 0 for i ≠ j. If each ui is normalized, the resulting set forms an orthonormal basis.

The Gram-Schmidt process is the standard algorithm for orthogonalization; variants such as the modified Gram-Schmidt improve numerical stability. Other approaches, notably Householder reflections, are used to perform QR factorization, which relies on orthogonalization.

Applications span solving least squares problems, constructing bases for subspaces, and enabling efficient computations in numerical linear algebra, signal processing, statistics, and data compression, where decorrelation or whitening is desirable.

Terminology varies by language: ortogonaliset or related forms appear in several European languages, while English uses orthogonalization or orthogonalisation. The underlying idea remains the same: produce an orthogonal (and optionally orthonormal) set from an original collection of vectors.