Login
Login

biorthogonal system

In mathematics, a biorthogonal system is a pair of indexed families of vectors

\tilde v_i \text{ in } E \text{ and } \tilde u_i \text{ in } F

such that

\left\langle\tilde v_i , \tilde u_j\right\rangle = \delta_{i,j},

where E and F form a pair of topological vector spaces that are in duality, \langle \,\cdot, \cdot\, \rangle is a bilinear mapping and \delta_{i,j} is the Kronecker delta.

An example is the pair of sets of respectively left and right eigenvectors of a matrix, indexed by eigenvalue, if the eigenvalues are distinct.{{cite book|last1=Bhushan|first1=Datta, Kanti|title=Matrix And Linear Algebra, Edition 2: AIDED WITH MATLAB|date=2008|publisher=PHI Learning Pvt. Ltd.|isbn=9788120336186|page=239|url=https://books.google.com/books?id=4kId0byq_ooC&q=biorthogonal+system+eigenvector&pg=PA239|language=en}}

A biorthogonal system in which E = F and \tilde v_i = \tilde u_i is an orthonormal system.

Projection

Related to a biorthogonal system is the projection

P := \sum_{i \in I} \tilde u_i \otimes \tilde v_i,

where (u \otimes v) (x) := u \langle v, x \rangle; its image is the linear span of \left\{\tilde u_i: i \in I\right\}, and the kernel is \left\{\left\langle \tilde v_i, \cdot \right\rangle = 0 : i \in I\right\}.

Construction

Given a possibly non-orthogonal set of vectors \mathbf{u} = \left(u_i\right) and \mathbf{v} = \left(v_i\right) the projection related is

P = \sum_{i,j} u_i \left(\langle\mathbf{v}, \mathbf{u}\rangle^{-1}\right)_{j,i} \otimes v_j,

where \langle\mathbf{v},\mathbf{u}\rangle is the matrix with entries \left(\langle\mathbf{v}, \mathbf{u}\rangle\right)_{i,j} = \left\langle v_i, u_j\right\rangle.

  • \tilde u_i := (I - P) u_i, and \tilde v_i := (I - P)^* v_i then is a biorthogonal system.

See also

  • {{annotated link|Dual basis}}
  • {{annotated link|Dual space}}
  • {{annotated link|Dual pair}}
  • {{annotated link|Orthogonality}}
  • {{annotated link|Orthogonalization}}

References

{{reflist}}

{{reflist|group=note}}

  • Jean Dieudonné, On biorthogonal systems Michigan Math. J. 2 (1953), no. 1, 7–20 [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.mmj/1028989861]

{{Duality and spaces of linear maps}}

{{Functional analysis}}

Category:Topological vector spaces