Biorthogonal system

Template:Short description 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 $$⟨{\tilde{v}}_i,{\tilde{u}}_j⟩={\delta }_{i,j},$$ where $E$ and $F$ form a pair of topological vector spaces that are in duality, $⟨\cdot ,\cdot ⟩$ 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 $(u\otimes v)(x):=u⟨v,x⟩;$ its image is the linear span of $\{{\tilde{u}}_i:i\in I\},$ and the kernel is $\{⟨{\tilde{v}}_i,\cdot ⟩=0:i\in I\}.$

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 $$P=\sum _{i,j}{u}_i{(⟨\mathbf{𝐯},\mathbf{𝐮}{⟩}^{-1})}_{j,i}\otimes v_j,$$ where $⟨\mathbf{𝐯},\mathbf{𝐮}⟩$ is the matrix with entries ${(⟨\mathbf{𝐯},\mathbf{𝐮}⟩)}_{i,j}=⟨v_i,u_j⟩.$

• ${\tilde{u}}_i:=(I-P)u_i,$ 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 [1]