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In mathematics , a biorthogonal system is a pair of indexed families of vectors
v
~
i
{\displaystyle {\tilde {v}}_{i}}
in E and
u
~
i
{\displaystyle {\tilde {u}}_{i}}
in F
such that
⟨
v
~
i
,
u
~
j
⟩
=
δ
i
,
j
,
{\displaystyle \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 , ⟨·,·⟩ is a bilinear mapping and
δ
i
,
j
{\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 E = F and
v
~
i
=
u
~
i
{\displaystyle {\tilde {v}}_{i}={\tilde {u}}_{i}}
is an orthonormal system .
Projection
Related to a biorthogonal system is the projection
P
:=
∑
i
∈
I
u
~
i
⊗
v
~
i
{\displaystyle P:=\sum _{i\in I}{\tilde {u}}_{i}\otimes {\tilde {v}}_{i}}
,
where
(
u
⊗
v
)
(
x
)
:=
u
⟨
v
,
x
⟩
{\displaystyle \left(u\otimes v\right)(x):=u\langle v,x\rangle }
; its image is the linear span of
{
u
~
i
:
i
∈
I
}
{\displaystyle \left\{{\tilde {u}}_{i}:i\in I\right\}}
, and the kernel is
{
⟨
v
~
i
,
⋅
⟩
=
0
:
i
∈
I
}
{\displaystyle \left\{\left\langle {\tilde {v}}_{i},\cdot \right\rangle =0:i\in I\right\}}
.
Construction
Given a possibly non-orthogonal set of vectors
u
=
(
u
i
)
{\displaystyle \mathbf {u} =(u_{i})}
and
v
=
(
v
i
)
{\displaystyle \mathbf {v} =\left(v_{i}\right)}
the projection related is
P
=
∑
i
,
j
u
i
(
⟨
v
,
u
⟩
−
1
)
j
,
i
⊗
v
j
{\displaystyle P=\sum _{i,j}u_{i}\left(\langle \mathbf {v} ,\mathbf {u} \rangle ^{-1}\right)_{j,i}\otimes v_{j}}
,
where
⟨
v
,
u
⟩
{\displaystyle \langle \mathbf {v} ,\mathbf {u} \rangle }
is the matrix with entries
(
⟨
v
,
u
⟩
)
i
,
j
=
⟨
v
i
,
u
j
⟩
{\displaystyle \left(\langle \mathbf {v} ,\mathbf {u} \rangle \right)_{i,j}=\left\langle v_{i},u_{j}\right\rangle }
.
u
~
i
:=
(
I
−
P
)
u
i
{\displaystyle {\tilde {u}}_{i}:=(I-P)u_{i}}
, and
v
~
i
:=
(
I
−
P
)
∗
v
i
{\displaystyle {\tilde {v}}_{i}:=\left(I-P\right)^{*}v_{i}}
then is a biorthogonal system.
See also
References
Jean Dieudonné, On biorthogonal systems Michigan Math. J. 2 (1953), no. 1, 7–20 [1]