Projection (linear algebra)

The linear mapping T is the projection along k onto m. All points in the image m (e.g. w ) are mapped from T onto itself (e.g. Tw ).

In mathematics , a projection or projector is a special linear mapping ( endomorphism ) over a vector space that leaves all vectors in its image (a subspace of ) unchanged. ${\displaystyle V}$ ${\displaystyle V}$

If a base of is chosen appropriately , the projection sets some components of a vector to zero and keeps the others. This clearly justifies the term projection, such as the illustration of a house in a two-dimensional floor plan. ${\displaystyle V}$

definition

Let be a vector space . A vector space endomorphism is called a projection if it is idempotent , i.e. if it holds. ${\displaystyle V}$ ${\displaystyle P\colon V\to V}$ ${\displaystyle P\circ P=P}$

properties

A projection can only have the numbers 0 and 1 as eigenvalues . The eigenspaces are

• ${\displaystyle \ker P}$( Core of ) to the eigenvalue 0 and ${\displaystyle P}$
• ${\displaystyle \operatorname {im} P}$( Image from ) to the eigenvalue 1. ${\displaystyle P}$

The total space is the direct sum of these two subspaces :

${\displaystyle V=\ker P\oplus \operatorname {im} P}$

The illustration is clearly speaking a parallel projection along . ${\displaystyle P}$ ${\displaystyle \operatorname {im} P}$ ${\displaystyle \ker P}$

If there is a projection, then there is also a projection, and the following applies: ${\displaystyle P}$ ${\displaystyle \operatorname {id} -P}$

${\displaystyle \ker P=\operatorname {im} (\operatorname {id} -P)}$

${\displaystyle \operatorname {im} P=\ker(\operatorname {id} -P)}$

Projections and complements

If there is a vector space and a subspace, then in general there are many projections on , that is, projections whose image is. If there is a projection with an image , then there is a complement to in . ${\displaystyle V}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle P}$ ${\displaystyle U}$ ${\displaystyle \ker P}$ ${\displaystyle U}$ ${\displaystyle V}$

Conversely, if a complement of in , so , then each can as a sum with clearly specified and represent. The endomorphism of , which assigns what belongs to each , is a projection with an image and a core . Projections and decompositions into complementary subspaces correspond to one another. ${\displaystyle W}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle V=U\oplus W}$ ${\displaystyle v\in V}$ ${\displaystyle v=u+w}$ ${\displaystyle u\in U}$ ${\displaystyle w\in W}$ ${\displaystyle V}$ ${\displaystyle v}$ ${\displaystyle u}$ ${\displaystyle U}$ ${\displaystyle W}$

Orthogonal projection

Orthogonal decomposition of a vector into a part in a plane and a part in the orthogonal complement of the plane ${\displaystyle v}$ ${\displaystyle P(v)=u}$ ${\displaystyle U}$ ${\displaystyle v-P(v)=u^{\perp }}$ ${\displaystyle U^{\perp }}$

→ Main article : Orthogonal projection

If a finite-dimensional real or complex vector space with a scalar product , there is a projection along the orthogonal complement of for every sub-vector space , which is called “orthogonal projection onto ”. It is the uniquely determined linear mapping with the property that for all ${\displaystyle V}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle P\colon V\to V}$ ${\displaystyle v\in V}$

• ${\displaystyle P(v)\in U}$ and
• ${\displaystyle v-P(v)\perp U}$

applies.

If an infinite-dimensional Hilbert space , then this statement with the projection theorem applies accordingly to closed sub-vector spaces . In this case, continuous can be chosen. ${\displaystyle V}$ ${\displaystyle U}$ ${\displaystyle P}$

Examples

As simple examples can be for each vector space the identity and the figure for state as trivial projections (which can be represented by the unity or zero matrix). ${\displaystyle V}$ ${\displaystyle Id:v\mapsto v}$ ${\displaystyle N\colon v\mapsto 0}$ ${\displaystyle v\in V}$

Let it be the mapping of the plane in itself that is passed through the matrix ${\displaystyle P}$ ${\displaystyle \mathbb {R} ^{2}}$

${\displaystyle P={\begin{pmatrix}1&0\\0&0\end{pmatrix}}}$

is described. It projects a vector onto , i.e. orthogonally onto the x-axis. The eigen-space for eigenvalue , i.e. the core, is spanned by , the eigen-space for eigenvalue , i.e. the image, is spanned by. The projector is the orthogonal projection on the y-axis. ${\displaystyle {\tbinom {x}{y}}}$ ${\displaystyle {\tbinom {x}{0}}}$ ${\displaystyle 0}$ ${\displaystyle {\tbinom {0}{1}}}$ ${\displaystyle 1}$ ${\displaystyle {\tbinom {1}{0}}}$ ${\displaystyle \operatorname {id} -P}$

In contrast, for example, is through the matrix

${\displaystyle P={\begin{pmatrix}1&-1\\0&0\end{pmatrix}}}$

The described image of the plane is also due to a projection, but not an orthogonal projection. Its image is again the x-axis, but its core is the straight line with the equation . ${\displaystyle P^{2}=P}$ ${\displaystyle y=x}$

application

In quantum mechanics , in connection with the measurement process , one speaks of a projection of the state vector ψ, the precise interpretation being described below:

• As a measurement result , only one of i comes. A. question an infinite number of so-called eigenvalues ​​of the observed observables (ie the assigned self-adjoint operator in the state space of the system, the so-called Hilbert space ). The selection is made randomly ( Copenhagen interpretation ) with a certain probability, which is not required here.
• The calculation of the probability for an eigenvalue (measurement result) takes place, among other things, with the aid of the projection onto its eigenspace .

The totality of the projection operators obtained in this way is “complete” for a given measurand and results in the so-called spectral representation of the observables.

swell

• Gerd Fischer : Linear Algebra. Vieweg-Verlag, ISBN 3-528-03217-0 .
• Dirk Werner : Functional Analysis. 6th, corrected edition, Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 , page 161.