Biorthogonal system
In mathematics, a biorthogonal system is a pair of indexed families of vectors <math display=block>\tilde v_i \text{ in } E \text{ and } \tilde u_i \text{ in } F</math> such that <math display=block>\left\langle\tilde v_i , \tilde u_j\right\rangle = \delta_{i,j},</math> where <math>E</math> and <math>F</math> form a pair of topological vector spaces that are in duality, <math>\langle \,\cdot, \cdot\, \rangle</math> is a bilinear mapping and <math>\delta_{i,j}</math> 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.<ref name="Bhushan">Template:Cite book</ref>
A biorthogonal system in which <math>E = F</math> and <math>\tilde v_i = \tilde u_i</math> is an orthonormal system.
ProjectionEdit
Related to a biorthogonal system is the projection <math display=block>P := \sum_{i \in I} \tilde u_i \otimes \tilde v_i,</math> where <math>(u \otimes v) (x) := u \langle v, x \rangle;</math> its image is the linear span of <math>\left\{\tilde u_i: i \in I\right\},</math> and the kernel is <math>\left\{\left\langle \tilde v_i, \cdot \right\rangle = 0 : i \in I\right\}.</math>
ConstructionEdit
Given a possibly non-orthogonal set of vectors <math>\mathbf{u} = \left(u_i\right)</math> and <math>\mathbf{v} = \left(v_i\right)</math> the projection related is <math display=block>P = \sum_{i,j} u_i \left(\langle\mathbf{v}, \mathbf{u}\rangle^{-1}\right)_{j,i} \otimes v_j,</math> where <math> \langle\mathbf{v},\mathbf{u}\rangle </math> is the matrix with entries <math>\left(\langle\mathbf{v}, \mathbf{u}\rangle\right)_{i,j} = \left\langle v_i, u_j\right\rangle.</math>
- <math>\tilde u_i := (I - P) u_i,</math> and <math>\tilde v_i := (I - P)^* v_i</math> then is a biorthogonal system.
See alsoEdit
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ReferencesEdit
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- Jean Dieudonné, On biorthogonal systems Michigan Math. J. 2 (1953), no. 1, 7–20 [1]
Template:Duality and spaces of linear maps Template:Functional analysis