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13강. 일반최소제곱법과 QR 분할
Orthogonal Basis & Gram-Schmitt
are orthogonal
=> Left-inverse
=>
- Rotation matrix
- Permutation matrix
->
Geometrically, an orthogonal is the product of a rotation and a reflection
Projection reduces the length of a vector, But orthonormal matrix preserves angles and lengths
-> length conservation
-> inner product or angle conservation
- For any vector
=>
<=>
- 1-D projection onto a line
=
for
-> : projection of of
- => , for square cases The rows are also orthonormal
Rectangular matrix with Orthonormal columns
for m < n ; least square cases
: normal equation for least squares
: ()
는 Left-inverse이다.
m>n일 때는 가 로 나오지 않음.
Gram-Schmitt Orthogonalization
-> find the orthonormal basis given independent vectors
이는 다음과 같다.
아래는 공식화 한 것이다.
Factorization
- linear combination of each column vector
Upper triangular matrix 모양이다
이를 이라고 해서 QR분해라고한다.
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