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.

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

• Dual basis – Linear algebra concept
• Dual space – In mathematics, vector space of linear forms
• Dual pair
• Orthogonality – Various meanings of the terms
• Orthogonalization

References

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