Biorthogonal system

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

\tilde v_i in E and \tilde u_i in F

such that

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

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

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

An example is the pair of sets of respectively left and right eigenvectors of a matrix, indexed by eigenvalue.[citation needed]

Projection[edit]

Related to a biorthogonal system is the projection

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

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

Construction[edit]

Given a possibly non-orthogonal set of vectors \mathbf{u}= (u_i) and \mathbf{v}= (v_i) 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}= \langle v_i, u_j\rangle .

  • \tilde u_i:= (I-P) u_i, and \tilde v_i:= \left(I-P \right)^* v_i then is an orthogonal system.

See also[edit]

References[edit]

  • Jean Dieudonné, On biorthogonal systems Michigan Math. J. 2 (1953), no. 1, 7–20 [1]