biorthogonal system

In mathematics, a biorthogonal system is a pair of indexed families of vectors
$\tilde v_i \text E \text \tilde u_i \text F$
such that
$\left\langle\tilde v_i , \tilde u_j\right\rangle = \delta_,$
where $E$ and $F$ form a pair of topological vector spaces that are in duality, $\langle \,\cdot, \cdot\, \rangle$ is a bilinear mapping and $\delta_$ 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.

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
$P := \sum_ \tilde u_i \otimes \tilde v_i,$
where $(u \otimes v) (x) := u \langle v, x \rangle;$ its image is the linear span of $\left\,$ and the kernel is $\left\.$

Construction

Given a possibly non-orthogonal set of vectors $\mathbf = \left(u_i\right)$ and $\mathbf = \left(v_i\right)$ the projection related is
$P = \sum_ u_i \left(\langle\mathbf, \mathbf\rangle^\right)_ \otimes v_j,$
where $\langle\mathbf,\mathbf\rangle$ is the matrix with entries $\left(\langle\mathbf, \mathbf\rangle\right)_ = \left\langle v_i, u_j\right\rangle.$
* $\tilde u_i := (I - P) u_i,$ and $\tilde v_i := (I - P)^* v_i$ then is a biorthogonal system.

See also

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References

* Jean Dieudonné, ''On biorthogonal systems'' Michigan Math. J. 2 (1953), no. 1, 7–20

{{Functional analysis

Topological vector spaces