Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.[1][2][3] For example, the standard basis for a Euclidean space is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for arises in this fashion.

For a general inner product space an orthonormal basis can be used to define normalized orthogonal coordinates on Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of under dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram–Schmidt process.

In functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces.[4] Given a pre-Hilbert space an orthonormal basis for is an orthonormal set of vectors with the property that every vector in can be written as an infinite linear combination of the vectors in the basis. In this case, the orthonormal basis is sometimes called a Hilbert basis for Note that an orthonormal basis in this sense is not generally a Hamel basis, since infinite linear combinations are required. Specifically, the linear span of the basis must be dense in but it may not be the entire space.

If we go on to Hilbert spaces, a non-orthonormal set of vectors having the same linear span as an orthonormal basis may not be a basis at all. For instance, any square-integrable function on the interval can be expressed (almost everywhere) as an infinite sum of Legendre polynomials (an orthonormal basis), but not necessarily as an infinite sum of the monomials

A different generalisation is to pseudo-inner product spaces, finite-dimensional vector spaces equipped with a non-degenerate symmetric bilinear form known as the metric tensor. In such a basis, the metric takes the form with positive ones and negative ones.

ExamplesEdit

  • For  , the set of vectors   is called the standard basis and forms an orthonormal basis of   with respect to the standard dot product. Note that both the standard basis and standard dot product rely on viewing   as the Cartesian product  
    Proof: A straightforward computation shows that the inner products of these vectors equals zero,   and that each of their magnitudes equals one,   This means that   is an orthonormal set. All vectors   can be expressed as a sum of the basis vectors scaled
     
    so   spans   and hence must be a basis. It may also be shown that the standard basis rotated about an axis through the origin or reflected in a plane through the origin also forms an orthonormal basis of  .
  • For  , the standard basis and inner product are similarly defined. Any other orthonormal basis is related to the standard basis by an orthogonal transformation in the group O(n).
  • For pseudo-Euclidean space  , an orthogonal basis   with metric   instead satisfies   if  ,   if  , and   if  . Any two orthonormal bases are related by a pseudo-orthogonal transformation. In the case  , these are Lorentz transformations.
  • The set   with   where   denotes the exponential function, forms an orthonormal basis of the space of functions with finite Lebesgue integrals,   with respect to the 2-norm. This is fundamental to the study of Fourier series.
  • The set   with   if   and   otherwise forms an orthonormal basis of  
  • Eigenfunctions of a Sturm–Liouville eigenproblem.
  • The column vectors of an orthogonal matrix form an orthonormal set.

Basic formulaEdit

If   is an orthogonal basis of   then every element   may be written as

 

When   is orthonormal, this simplifies to

 
and the square of the norm of   can be given by
 

Even if   is uncountable, only countably many terms in this sum will be non-zero, and the expression is therefore well-defined. This sum is also called the Fourier expansion of   and the formula is usually known as Parseval's identity.

If   is an orthonormal basis of   then   is isomorphic to   in the following sense: there exists a bijective linear map  such that

 

Incomplete orthogonal setsEdit

Given a Hilbert space   and a set   of mutually orthogonal vectors in   we can take the smallest closed linear subspace   of   containing   Then   will be an orthogonal basis of   which may of course be smaller than   itself, being an incomplete orthogonal set, or be   when it is a complete orthogonal set.

ExistenceEdit

Using Zorn's lemma and the Gram–Schmidt process (or more simply well-ordering and transfinite recursion), one can show that every Hilbert space admits an orthonormal basis;[5] furthermore, any two orthonormal bases of the same space have the same cardinality (this can be proven in a manner akin to that of the proof of the usual dimension theorem for vector spaces, with separate cases depending on whether the larger basis candidate is countable or not). A Hilbert space is separable if and only if it admits a countable orthonormal basis. (One can prove this last statement without using the axiom of choice.)

Choice of basis as a choice of isomorphismEdit

For concreteness we discuss orthonormal bases for a real,   dimensional vector space   with a positive definite symmetric bilinear form  .

One way to view an orthonormal basis with respect to   is as a set of vectors  , which allow us to write   for  , and   or  . With respect to this basis, the components of   are particularly simple:  

We can now view the basis as a map   which is an isomorphism of inner product spaces: to make this more explicit we can write

 

Explicitly we can write   where   is the dual basis element to  .

The inverse is a component map

 

These definitions make it manifest that there is a bijection

 

The space of isomorphisms admits actions of orthogonal groups at either the   side or the   side. For concreteness we fix the isomorphisms to point in the direction  , and consider the space of such maps,  .

This space admits a left action by the group of isometries of  , that is,   such that  , with the action given by composition:  

This space also admits a right action by the group of isometries of  , that is,  , with the action again given by composition:  .

As a principal homogeneous spaceEdit

The set of orthonormal bases for   with the standard inner product is a principal homogeneous space or G-torsor for the orthogonal group   and is called the Stiefel manifold   of orthonormal  -frames.[6]

In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given the space of orthonormal bases, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group. Concretely, a linear map is determined by where it sends a given basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis.

The other Stiefel manifolds   for   of incomplete orthonormal bases (orthonormal  -frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any  -frame can be taken to any other  -frame by an orthogonal map, but this map is not uniquely determined.

  • The set of orthonormal bases for   is a G-torsor for  .
  • The set of orthonormal bases for   is a G-torsor for  .
  • The set of orthonormal bases for   is a G-torsor for  .
  • The set of right-handed orthonormal bases for   is a G-torsor for  

See alsoEdit

ReferencesEdit

  1. ^ Lay, David C. (2006). Linear Algebra and Its Applications (3rd ed.). Addison–Wesley. ISBN 0-321-28713-4.
  2. ^ Strang, Gilbert (2006). Linear Algebra and Its Applications (4th ed.). Brooks Cole. ISBN 0-03-010567-6.
  3. ^ Axler, Sheldon (2002). Linear Algebra Done Right (2nd ed.). Springer. ISBN 0-387-98258-2.
  4. ^ Rudin, Walter (1987). Real & Complex Analysis. McGraw-Hill. ISBN 0-07-054234-1.
  5. ^ Linear Functional Analysis Authors: Rynne, Bryan, Youngson, M.A. page 79
  6. ^ "CU Faculty". engfac.cooper.edu. Retrieved 2021-04-15.

External linksEdit

  • This Stack Exchange Post discusses why the set of Dirac Delta functions is not a basis of L2([0,1]).