Biorthogonal system

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.

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

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 $\left(u\otimes v\right)(x):=u\langle v,x\rangle$; 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

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 a biorthogonal system.

See also

• Dual basis
• Dual space
• Dual pair
• Orthogonality
• Orthogonalization

References

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