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| '''Gram-Schmidt orthonormalization''' is a process for making vectors into orthonormal vectors. | '''Gram-Schmidt orthonormalization''' is a process for transforming a vectors into [[Calculus/UnitVector|unit vectors]]. |
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| Two vectors ''a'' and ''b'' can be orthonormalized into ''A'' and ''B''. | Two vectors ''a'' and ''b'' can be orthonormalized into ''A'' and ''B''. This means... 1. that they are made [[Calculus/Orthogonality|orthogonal]] to each other by removing any components of one from the other. 2. that they are normalized to a unit [[Calculus/Distance|distance]] of 1. |
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| [[Calculus/Orthogonality|Orthogonality]] is a property of two vectors, not one. Therefore ''a'' needs no transformation and becomes ''A''. | These are accomplished in discrete steps. The first is to enforce orthogonality. But orthogonality is a property of two vectors, not one. Therefore ''a'' needs no transformation to become ''A''. |
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| The orthogonal vectors are then normalized by scaling to their [[Calculus/Distance#Euclidean_distance|Euclidean distances]], as ''A/||A||'' and ''B/||B||''. |
The final step is to normalize the orthogonal vectors by their own [[Calculus/Distance#Euclidean_distance|distance]], as in ''A/||A||'' and ''B/||B||''. |
Orthonormalization
Gram-Schmidt orthonormalization is a process for transforming a vectors into unit vectors.
Contents
Vectors
Two vectors a and b can be orthonormalized into A and B. This means...
that they are made orthogonal to each other by removing any components of one from the other.
that they are normalized to a unit distance of 1.
These are accomplished in discrete steps. The first is to enforce orthogonality. But orthogonality is a property of two vectors, not one. Therefore a needs no transformation to become A.
The process of transforming b into 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. 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 = (ATb)/(ATA). Therefore, B = b - A (ATb)/(ATA).
To transform another vector c into being orthogonal to both A and B, apply the same process for each component: C = c - A (ATc)/(ATA) - B (BTc)/(BTB).
The final step is to normalize the orthogonal vectors by their own distance, as in 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 v1 and v2 in V. The process yields u1 and u2 in U. Then the columns are normalized into Q like q1 = u1/||u1||.
To re-emphasize, this is a linear combination generalized as A = QR, and does not change the column space of A.
Note that Q is a matrix with orthonormal columns; it must also be square to be called an orthogonal matrix.