Biorthogonal system

In mathematics, a biorthogonal system is a pair of indexed families of vectors $${\displaystyle {\tilde {v}}_{i}{\text{ in }}E{\text{ and }}{\tilde {u}}_{i}{\text{ in }}F}$$ such that $${\displaystyle \left\langle {\tilde {v}}_{i},{\tilde {u}}_{j}\right\rangle =\delta _{i,j},}$$ where ${\displaystyle E}$ and ${\displaystyle F}$ form a pair of topological vector spaces that are in duality, ${\displaystyle \langle \,\cdot ,\cdot \,\rangle }$ is a bilinear mapping and ${\displaystyle \delta _{i,j}}$ 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.^[1]

A biorthogonal system in which ${\displaystyle E=F}$ and ${\displaystyle {\tilde {v}}_{i}={\tilde {u}}_{i}}$ is an orthonormal system.

Projection

Related to a biorthogonal system is the projection $${\displaystyle P:=\sum _{i\in I}{\tilde {u}}_{i}\otimes {\tilde {v}}_{i},}$$ where ${\displaystyle (u\otimes v)(x):=u\langle v,x\rangle$ ;} its image is the linear span of ${\displaystyle \left\{{\tilde {u}}_{i}:i\in I\right\},}$ and the kernel is ${\displaystyle \left\{\left\langle {\tilde {v}}_{i},\cdot \right\rangle =0:i\in I\right\}.}$

Construction

Given a possibly non-orthogonal set of vectors ${\displaystyle \mathbf {u} =\left(u_{i}\right)}$ and ${\displaystyle \mathbf {v} =\left(v_{i}\right)}$ the projection related is $${\displaystyle P=\sum _{i,j}u_{i}\left(\langle \mathbf {v} ,\mathbf {u} \rangle ^{-1}\right)_{j,i}\otimes v_{j},}$$ where ${\displaystyle \langle \mathbf {v} ,\mathbf {u} \rangle }$ is the matrix with entries ${\displaystyle \left(\langle \mathbf {v} ,\mathbf {u} \rangle \right)_{i,j}=\left\langle v_{i},u_{j}\right\rangle .}$

• ${\displaystyle {\tilde {u}}_{i}:=(I-P)u_{i},}$ and ${\displaystyle {\tilde {v}}_{i}:=(I-P)^{*}v_{i}}$ 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