orthogonalisoida

Orthogonalisoida is a mathematical term used in Finnish to describe the process of converting a set of vectors into an orthogonal set that spans the same subspace. In more common terms, this is called orthogonalization. An orthogonal set satisfies ⟨vi, vj⟩ = 0 for i ≠ j, and an orthonormal set additionally has unit length for each vector. The concept applies to inner product spaces over real or complex fields.

The standard method is the Gram–Schmidt process. Given vectors v1, v2, ..., vk, one constructs u1 = v1 and for i > 1 defines ui = vi − ∑_{j=1}^{i−1} proj_{uj}(vi), where the projection uses the inner product. In real spaces, proj_{uj}(vi) = (vi·uj / uj·uj) uj. In complex spaces, the inner product involves complex conjugation. After obtaining {u1, ..., uk}, one may normalize to produce an orthonormal set. Other approaches to orthogonalization include QR decomposition A = QR, computed with Householder reflections or Givens rotations.

Properties and applications: If the original vectors are linearly independent, the orthogonalized set is a basis for the same subspace; if not, it spans the same subspace as the original vectors. Orthogonalization is used in solving least squares problems, in numerical linear algebra for stable basis construction, and in algorithms that require orthogonal coordinates. Classical Gram–Schmidt can be numerically unstable for nearly dependent vectors; modified Gram–Schmidt or reorthogonalization are common remedies.

In analysis, orthogonalization also appears in function spaces, yielding orthogonal polynomials (such as Legendre or Chebyshev) and in Fourier-type expansions where basis functions are made orthogonal with respect to a weight. The Finnish term orthogonalisoida appears in mathematical texts to describe these procedures.