|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[mathematics]], a '''biorthogonal system''' is a pair of [[indexed family|indexed families]] of vectors
| | I'm Mattie and I live with my husband and our two children in Froidmont, in the WHT south part. My hobbies are Radio-Controlled Car Racing, Skiing and Bus spotting.<br><br>My blog: [http://transmojal.com.mx/punbb/profile.php?id=19431 remove paint from wood] |
| :<math>\tilde v_i</math> in {{mvar|E}} and <math>\tilde u_i</math> in {{mvar|F}}
| |
| such that
| |
| :<math> \langle\tilde v_i , \tilde u_j\rangle = \delta_{i,j} ,</math>
| |
| where ''E'' and ''F'' form a pair of [[topological vector space]]s that are in [[dual space|duality]], {{math|{{langle}},{{rangle}}}} is a [[bilinear mapping]] and <math>\delta_{i,j}</math> is the [[Kronecker delta]].
| |
| | |
| A biorthogonal system in which {{math|1={{var|E}} = {{var|F}}}} and <math>\tilde v_i = \tilde u_i</math> is an [[Orthogonal basis|orthonormal system]].
| |
| | |
| An example is the pair of sets of respectively left and right eigenvectors of a matrix, indexed by eigenvalue.{{cn|date=January 2013}}
| |
| | |
| ==Projection==
| |
| Related to a biorthogonal system is the projection
| |
| :<math>P:= \sum_{i \in I} \tilde u_i \otimes \tilde v_i </math>,
| |
| where <math>\left( u \otimes v\right) (x):= u \langle v, x\rangle</math>; its image is the [[linear span]] of <math>\{\tilde u_i: i \in I\}</math>, and the
| |
| [[kernel (algebra)|kernel]] is <math>\{\langle\tilde v_i, \cdot\rangle = 0: i \in I \}</math>.
| |
| | |
| ==Construction==
| |
| Given a possibly non-orthogonal set of vectors <math>\mathbf{u}= (u_i)</math> and <math>\mathbf{v}= (v_i)</math> the projection related is
| |
| :<math>P= \sum_{i,j} u_i \left( \langle\mathbf{v}, \mathbf{u}\rangle^{-1}\right)_{j,i} \otimes v_j</math>, | |
| where <math> \langle\mathbf{v},\mathbf{u}\rangle </math> is the matrix with entries <math> \left(\langle\mathbf{v},\mathbf{u}\rangle\right)_{i,j}= \langle v_i, u_j\rangle </math>.
| |
| * <math>\tilde u_i:= (I-P) u_i</math>, and <math>\tilde v_i:= \left(I-P \right)^* v_i</math> then is an orthogonal system.
| |
| | |
| ==See also==
| |
| * [[Dual space]]
| |
| * [[Dual pair]]
| |
| * [[Orthogonality]]
| |
| * [[Orthogonalization]]
| |
| | |
| ==References==
| |
| | |
| *Jean Dieudonné, ''On biorthogonal systems'' Michigan Math. J. 2 (1953), no. 1, 7–20 [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.mmj/1028989861]
| |
| | |
| [[Category:Topological vector spaces]]
| |
I'm Mattie and I live with my husband and our two children in Froidmont, in the WHT south part. My hobbies are Radio-Controlled Car Racing, Skiing and Bus spotting.
My blog: remove paint from wood