Biorthogonal system

In mathematics, a biorthogonal system is a pair of indexed families of vectors [math]\displaystyle{ \tilde v_i \text{ in } E \text{ and } \tilde u_i \text{ in } F }[/math] such that [math]\displaystyle{ \left\langle\tilde v_i , \tilde u_j\right\rangle = \delta_{i,j}, }[/math] where [math]\displaystyle{ E }[/math] and [math]\displaystyle{ F }[/math] form a pair of topological vector spaces that are in duality, [math]\displaystyle{ \langle \,\cdot, \cdot\, \rangle }[/math] is a bilinear mapping and [math]\displaystyle{ \delta_{i,j} }[/math] 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 [math]\displaystyle{ E = F }[/math] and [math]\displaystyle{ \tilde v_i = \tilde u_i }[/math] is an orthonormal system.

Projection

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

Construction

Given a possibly non-orthogonal set of vectors [math]\displaystyle{ \mathbf{u} = \left(u_i\right) }[/math] and [math]\displaystyle{ \mathbf{v} = \left(v_i\right) }[/math] the projection related is [math]\displaystyle{ P = \sum_{i,j} u_i \left(\langle\mathbf{v}, \mathbf{u}\rangle^{-1}\right)_{j,i} \otimes v_j, }[/math] where [math]\displaystyle{ \langle\mathbf{v},\mathbf{u}\rangle }[/math] is the matrix with entries [math]\displaystyle{ \left(\langle\mathbf{v}, \mathbf{u}\rangle\right)_{i,j} = \left\langle v_i, u_j\right\rangle. }[/math]

• [math]\displaystyle{ \tilde u_i := (I - P) u_i, }[/math] and [math]\displaystyle{ \tilde v_i := (I - P)^* v_i }[/math] then is a biorthogonal system.

See also

• Dual basis – Linear algebra concept
• Dual space – In mathematics, vector space of linear forms
• Dual pair
• Orthogonality – Other name of perpendicularity and its generalizations
• Orthogonalization

References

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