Biorthogonal system

In mathematics, a biorthogonal system is a pair of indexed families of vectors

\tilde v_i

E

and

\tilde u_i

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.

A biorthogonal system in which $E = F$ and $\tilde v_i = \tilde u_i$ is an orthonormal system.

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

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 an orthogonal system.

References

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

Biorthogonal system

Projection

Construction

See also

References