How to evaluate Nonorthogonal basis?

  • Context: Graduate 
  • Thread starter Thread starter marshall.L
  • Start date Start date
  • Tags Tags
    Basis
Click For Summary
SUMMARY

This discussion focuses on evaluating the performance of nonorthogonal basis sets in Hilbert space. Users highlight that while orthonormal bases allow for straightforward coefficient extraction via dot products, nonorthogonal bases can also be assessed through reconstruction error, which measures their descriptive ability. The conversation emphasizes the need for clarity in defining evaluation metrics for basis vectors, particularly in nonorthogonal contexts.

PREREQUISITES
  • Understanding of Hilbert space concepts
  • Familiarity with basis vectors and their properties
  • Knowledge of reconstruction error as a performance metric
  • Basic linear algebra, including dot products
NEXT STEPS
  • Research methods for calculating reconstruction error in vector spaces
  • Explore the properties and applications of nonorthogonal bases
  • Learn about the implications of using orthonormal versus nonorthogonal bases in data representation
  • Study advanced topics in functional analysis related to Hilbert spaces
USEFUL FOR

Mathematicians, physicists, and data scientists interested in vector space theory, particularly those evaluating the performance of basis sets in Hilbert space.

marshall.L
Messages
2
Reaction score
0
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:
 
Physics news on Phys.org
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.
 
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:
 

Similar threads

Undergrad Non orthogonal basis and the lines of its coordinate grid
  • · Replies 9 ·
Replies
9
Views
4K
Undergrad Proof that if T is Hermitian, eigenvectors form an orthonormal basis
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad What is the difference between a complete basis and an overcomplete dictionary?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Consistent matrix index notation when dealing with change of basis
  • · Replies 12 ·
Replies
12
Views
3K
Graduate Shauder basis for Hilbert spaces
  • · Replies 0 ·
Replies
0
Views
780
Undergrad Why should a Fourier transform not be a change of basis?
  • · Replies 43 ·
2
Replies
43
Views
8K
Undergrad Basis of a Subspace of a Vector Space
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Spectral theorem for Hermitian matrices-- special cases
  • · Replies 2 ·
Replies
2
Views
3K
MHB Question about proof of the linear independence of a dual basis
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad How to Construct an Orthonormal Basis for a 2D Subspace in Linear Algebra?
  • · Replies 3 ·
Replies
3
Views
2K