그람-슈미트 직교정규화

노트

말뭉치

1. It's called the Gram-Schmidt process.
2. We have done the Gram-Schmidt process.
3. And this process of creating an orthonormal basis is called the Gram-Schmidt Process.
4. We can apply the Gram-Schmidt procedure to the basis and obtain an orthonormal set .
5. We now come to a fundamentally important algorithm, which is called the Gram-Schmidt orthogonalization procedure.
6. Apply the Gram-Schmidt procedure to this list to obtain an orthonormal list \((e_1,\ldots,e_n)\), which still spans \(V \) by construction.
7. Now apply the Gram-Schmidt procedure to obtain a new orthonormal basis \((e_1,\ldots,e_m,f_1,\ldots,f_k)\).
8. Gram-Schmidt process, or orthogonalisation, is a way to transform the vectors of the basis of a subspace from an arbitrary alignment to an orthonormal basis.
9. The Gram-Schmidt process takes those vectors and generates the same number of vectors organized as an orthonormal system.
10. It is often much simpler to perform calculations in an orthogonal basis, and the Gram-Schmidt process constructs an orthonormal (orthogonal and normalized) basis.
11. The modified Gram-Schmidt process being executed on three linearly independent, non-orthogonal vectors of a basis for R3.
12. The Gram–Schmidt process also applies to a linearly independent countably infinite sequence {v i } i .
13. If the Gram–Schmidt process is applied to a linearly dependent sequence, it outputs the 0 vector on the ith step, assuming that v i is a linear combination of v 1 , ..., v i−1 .
14. The Gram–Schmidt process can be stabilized by a small modification; this version is sometimes referred to as modified Gram-Schmidt or MGS.
15. Gram-Schmidt Orthogonalization has long been recognized for its numerical stability.
16. It has been argued in the literature for years that while Classical Gram-Schmidt Orthogonalization always requires reinforcement, Modified Gram-Schmidt never requires reorthogonalization.
17. The Gram-Schmidt Process produces an orthonormal basis for the subspace of Eucldiean n-space spanned by a finite set of vectors.
18. It carries out the Gram-Schmidt process directly by successively projecting each successive variable on the previous ones and subtracting (taking residuals).
19. Any arbitrary basis can be transformed to an orthonormal basis by a procedure known as Gram–Schmidt orthonormalization.
20. That's exactly what the Gram-Schmidt process is for, as we'll see in a second.
21. If we apply the Gram–Schmidt process to a well-ordered independent set whose closed linear span S S is not all of H H , we still get an orthonormal basis of the subspace S S .
22. If we apply the Gram–Schmidt process to a dependent set, then we will eventually run into a vector v v whose norm is zero, so we will not be able to take N ( v ) N(v) .
23. The modified Gram-Schmidt iteration uses orthogonal projectors in order ro make the process numerically more stable.
24. The Gram–Schmidt process is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space \( \mathbb{R}^n \) equipped with the standard inner product.
25. The method is named after a Danish actuary Jørgen Pedersen Gram (1850-1916) and a German mathematician Erhard Schmidt (1875-1959) but it appeared earlier in the work of Laplace and Cauchy.

메타데이터

위키데이터

• ID : Q475239

Spacy 패턴 목록

• [{'LOWER': 'gram'}, {'OP': '*'}, {'LOWER': 'schmidt'}, {'LEMMA': 'process'}]
• [{'LOWER': 'gram'}, {'OP': '*'}, {'LOWER': 'schmidt'}, {'LEMMA': 'process'}]
• [{'LOWER': 'gram'}, {'OP': '*'}, {'LOWER': 'schmidt'}, {'LEMMA': 'orthonormalization'}]
• [{'LOWER': 'gram'}, {'OP': '*'}, {'LOWER': 'schmidt'}, {'LEMMA': 'orthonormalization'}]
• [{'LOWER': 'gram'}, {'OP': '*'}, {'LOWER': 'schmidt'}, {'LEMMA': 'method'}]
• [{'LOWER': 'gram'}, {'OP': '*'}, {'LEMMA': 'Schmidt'}]