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}$${\ displaystyle vP (v) = u ^ {\ perp}}$${\ displaystyle U ^ {\ perp}}$

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
• ${\ displaystyle vP (v) \ perp U}$

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.