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| '''Gram-Schmidt orthonormalization''' is a process for making vectors into orthonormal vectors. It is generalized as '''''A''' = '''QR'''''. | '''Gram-Schmidt orthonormalization''' is a process for transforming a vectors into [[Calculus/UnitVector|unit vectors]]. |
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| [[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''. | Two vectors ''a'' and ''b'' can be orthonormalized into [[Calculus/UnitVector|unit vectors]] ''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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| 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)''. | 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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| Therefore, ''B = b - A (A^T^b)/(A^T^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''. [[LinearAlgebra/Projection|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)''. |
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| The orthogonal vectors are then normalized by scaling to their [[LinearAlgebra/Distance|Euclidean distances]], as ''A/||A||'' and ''B/||B||''. |
The final step is to normalize the orthogonal vectors by their own distance, as in ''A/||A||'' and ''B/||B||''. |
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| 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 generalized as '''''A''' = '''QR''''', and does not change the column space of '''''A'''''. |
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| To re-emphasize, this is a linear combination and does not change the column space. | Note that '''''Q''''' is a [[LinearAlgebra/Orthogonality|matrix with orthonormal columns]]; it must also be square to be called an '''orthogonal matrix'''. |
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 unit vectors 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.