= Orthonormalization = '''Gram-Schmidt orthonormalization''' is a process for making vectors into orthonormal vectors. It is generalized as '''''A''' = '''QR'''''. <> ---- == Vectors == [[LinearAlgebra/Orthogonality|Orthogonality]] is fundamentally about the relation between two vectors. So as the first point of reference, ''a'' needs no transformation. It is automatically denoted as the orthogonal vector ''A''. The process of transforming vector ''b'' into orthogonal vector ''B'' is simply the subtraction of all components of ''a'' from ''b''. This is a linear combination and does not change the column space of a system that includes both ''a'' and ''b''. [[LinearAlgebra/Projections#Vectors|Projections]] are a complimentary idea; ''p'' is the component of ''a'' that estimates ''b''. The process of '''orthonormalization''' is the same as computing projections but the error term ''e'' is the desired result. Recall that ''e = b - ax̂'' and ''x̂ = (A^T^b)/(A^T^A)''. Therefore, ''B = b - A (A^T^b)/(A^T^A)''. To transform another vector ''c'' into being orthogonal to ''both'' ''A'' and ''B'', apply the same process for each component: ''C = c - A (A^T^c)/(A^T^A) - B (B^T^c)/(B^T^B)''. The orthogonal vectors are then normalized by scaling to their [[LinearAlgebra/Distance|Euclidean distances]], as ''A/||A||'' and ''B/||B||''. ---- == Matrices == The process applied to vectors is also applicable to the columns in a matrix. Instead of vectors ''a'' and ''b'', use ''v,,1,,'' and ''v,,2,,'' in '''''V'''''. The process yields ''u,,1,,'' and ''u,,2,,'' in '''''U'''''. Then the columns are normalized into '''''Q''''' like ''q,,1,, = u,,1,,/||u,,1,,||''. Note that '''''Q''''' is a [[LinearAlgebra/Orthogonality#Matrices|matrix with orthonormal columns]], not necessarily an '''orthogonal matrix'''. To re-emphasize, this is a linear combination and does not change the column space. ---- CategoryRicottone