How to evaluate Nonorthogonal basis?

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In summary, the conversation discusses the evaluation of nonorthogonal basis sets, specifically in a Hibert space. The speaker is unsure of how to get started and is looking for a way to evaluate these bases in terms of their performance or advantage compared to orthogonal bases. The concept of dot product is mentioned as a property of orthonormal bases, but other than that, there is nothing special about orthogonal bases. The speaker also mentions using reconstruction error as a way to evaluate the descriptive ability of basis vectors, but is unsure if there are other measurement methods. They apologize for their lack of knowledge on the topic and their poor English.
  • #1
marshall.L
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hi all~
How to evaluate the performance of a set of nonorthogonal basis?
Like one in Hibert space which is most likely to be a nonorthongal set.
Does it have any advantage compared with orthogonal ones in any aspects?

i don't even know every to get started:confused:
 
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  • #2
I am not sure what you mean by "evaluate". "Orthonormal" bases have the nice property that the coefficient of the basis vector [itex]\vec{e}_i[/itex] in the expansion of [itex]\vec{v}[/itex] is just the dot product: [itex]\vec{e}_i\cdot\vec{v}[/itex].

Other than that, there is nothing special about orthogonal bases.
 
  • #3
HallsofIvy said:
I am not sure what you mean by "evaluate". "Orthonormal" bases have the nice property that the coefficient of the basis vector [itex]\vec{e}_i[/itex] in the expansion of [itex]\vec{v}[/itex] is just the dot product: [itex]\vec{e}_i\cdot\vec{v}[/itex].

Other than that, there is nothing special about orthogonal bases.


thx:)

i mean whether there is any kind of measurement which can be used to evaluate any aspect of a set of basis vectors.
i.e. we can use reconstruction error to evaluate the descriptive ability of a set of basis vectors.(The only way i know)

i havnt learned much on this aspect and i have searched on wikipedia for a long time with no progress.:frown:

i don't know whether i have made my question clear.
sry for my poor eng.:biggrin:
 

1. What is a Nonorthogonal basis?

A Nonorthogonal basis is a set of vectors that are not perpendicular to each other. Unlike an orthogonal basis, the vectors in a Nonorthogonal basis can have any angle between them.

2. Why is it important to evaluate Nonorthogonal basis?

Evaluating Nonorthogonal basis is important because it helps us determine the accuracy and efficiency of a mathematical model or algorithm. Nonorthogonal basis can introduce errors in calculations, so evaluating them helps us understand the reliability of our results.

3. How do you evaluate Nonorthogonal basis?

Nonorthogonal basis can be evaluated using various methods such as Gram-Schmidt orthogonalization, QR decomposition, or Singular Value Decomposition (SVD). These methods help us transform a Nonorthogonal basis into an orthogonal one, making it easier to analyze and interpret the data.

4. What are the advantages and disadvantages of Nonorthogonal basis?

The advantage of using Nonorthogonal basis is that it allows for a more flexible representation of data, as the vectors can have any angle between them. However, this also introduces potential errors in calculations and can make it more challenging to interpret the data. Additionally, Nonorthogonal basis can be computationally more demanding compared to an orthogonal basis.

5. Can Nonorthogonal basis be used in all types of data analysis?

Yes, Nonorthogonal basis can be used in many types of data analysis, such as dimensionality reduction, regression analysis, and machine learning. However, it is essential to evaluate the basis beforehand to understand its impact on the results and to choose the most appropriate method for analysis.

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