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Wikipedia / Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation from a vector space to itself (an endomorphism is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on pointss in the object.

Definitions

A projection.

has an inner product and is complete is a Hilbert space, the concept of orthogonality is called an orthogonal projection. A projection on a Hilbert space that is not orthogonal is called an oblique projection.

Projection matrix

The eigenvalues of a projection matrix must be 0 or 1.

Examples

Orthogonal projection

is an orthogonal projection onto the xy

The action of this matrix on an arbitrary vector

shows that the projection is an orthogonal projection.

Oblique projection

Via matrix multiplication is indeed a projection.

is orthogonal if and only if

Properties and classification

Idempotence

is idempotent).

Open map

Every projection is an open map onto its image, meaning that it maps each open set in the domain to an open set in the subspace topology of the image..

Complementarity of image and kernel

. Suppose the subspaces are the image and kernel has the following properties:

  1. is the identity operator

  2. We have a direct sum

The image and kernel of a projection are complementary.

Spectrum

In infinite-dimensional vector spaces, the spectrum Only 0 or 1 can be an eigenvalue is always a positive semi-definite matrix. In general, the corresponding eigenspaces.

If a projection is nontrivial it has minimal polynomial is diagonalizable.

Product of projections

The product of projections is not in general a projection, even if they are orthogonal. If two projections commute then their product is a projection, but the converse is false: the product of two non-commuting projections may or may not be a projection.

If two orthogonal projections commute then their product is an orthogonal projection. If the product of two orthogonal projections is an orthogonal projection, then the two orthogonal projections commute (more generally: two self-adjoint endomorphisms commute if and only if their product is self-adjoint).

Orthogonal projections

has an inner product and is complete (is a Hilbert space) the concept of orthogonality can be used. An orthogonal projection are orthogonal subspaces

A projection is orthogonal if and only if it is self-adjoint are orthogonal projections..

The existence of an orthogonal projection onto a closed subspace follows from the Hilbert projection theorem.

Properties and special cases

An orthogonal projection is a bounded operator in the vector space we have, by the Cauchy–Schwarz inequality.

For finite-dimensional complex or real vector spaces, the standard inner product.

Formulas

is a unit vector on the line, then the projection is given by the outer product is complex-valued, the transpose in the above equation is replaced by a Hermitian transpose). This operator leaves u, proving that it is indeed the orthogonal projection onto the line containing u. by the properties of the dot product of parallel and perpendicular vectors.

This formula can be generalized to orthogonal projections on a subspace of arbitrary dimension be an orthonormal basis. Then the projection is given by:

is the partial isometry that vanishes on the orthogonal complement is therefore the final space.

is a (not necessarily orthonormal) basis is the matrix with these vectors as columns, then the projection is:

is a "normalizing factor" that recovers the norm. For example, the rank.

In the general case, we can have an arbitrary positive definite

When the range space of the projection is generated by a frame stands for the Moore–Penrose pseudoinverse. This is just one of many ways to construct the projection operator.

is the null space),

non-singular

All these formulas also hold for complex inner product spaces, provided that the conjugate transpose is used instead of the transpose. Further details on sums of projectors can be found in Banerjee and Roy (2014). Also see Banerjee (2004) for application of sums of projectors in basic spherical trigonometry.

Oblique projections

The term oblique projections is sometimes used to refer to non-orthogonal projections. These projections are also used to represent spatial figures in two-dimensional drawings (see oblique projection), though not as frequently as orthogonal projections. Whereas calculating the fitted value of an ordinary least squares regression requires an orthogonal projection, calculating the fitted value of an instrumental variables regression requires an oblique projection.

A projection is defined by its kernel and the basis vectors used to characterize its range (which is a complement of the kernel). When these basis vectors are orthogonal to the kernel, then the projection is an orthogonal projection. When these basis vectors are not orthogonal to the kernel, the projection is an oblique projection, or just a projection.

A matrix representation formula for a nonzero projection operator

. It follows that the orthogonal complement

This expression generalizes the formula for orthogonal projections given above..

. In general, if the vector space is over complex number field, one then uses the Hermitian transpose. Recall that one can express the Moore–Penrose inverse.

Singular values

can be computed by an orthonormal basis be the orthogonal complement are: are equal, and thus that the matrix norm of the oblique projections are the same. However, the condition number, and is therefore not necessarily equal.

Finding projection with an inner product

where repeated indices are summed over (Einstein sum notation is the smallest distance (the orthogonal distance and is commonly used in areas such as machine learning.

Canonical forms

over a field is a diagonalizable matrix, since its minimal polynomial has the form

is the rank is the identity matrix is the zero matrix is the direct sum operator. If the vector space is complex and equipped with an inner product, then there is an orthonormal basis in which the matrix of P is

. The integers acts as an orthogonal projection (so that P-blocks correspond to the oblique components.

Projections on normed vector spaces

is a (not necessarily finite-dimensional) normed vector space is a Banach space.

.

However, in contrast to the finite-dimensional case, projections need not be continuous must be a closed subspace. Furthermore, the kernel of a continuous projection (in fact, a continuous linear operator in general) is closed. Thus a continuous into two complementary closed.

such that is continuous. This follows from the closed graph theorem. Suppose and is closed and , y, i.e. . Also, is closed and , which proves the claim.

, although for Hilbert spaces this can always be done by taking the orthogonal complement. For Banach spaces, a one-dimensional subspace always has a closed complementary subspace. This is an immediate consequence of Hahn–Banach theorem. By Hahn–Banach, there exists a bounded linear functional such that .

Applications and further considerations

Projections (orthogonal and otherwise) play a major role in algorithms for certain linear algebra problems:

As stated above, projections are a special case of idempotents. Analytically, orthogonal projections are non-commutative generalizations of characteristic functions. Idempotents are used in classifying, for instance, semisimple algebras, while measure theory begins with considering characteristic functions of measurable sets. Therefore, as one can imagine, projections are very often encountered in the context of operator algebras. In particular, a von Neumann algebra is generated by its complete lattice of projections.

Generalizations

be an isometry (compare Partial isometry); in particular it must be onto. The case of an orthogonal projection is when W is a subspace of V. In Riemannian geometry, this is used in the definition of a Riemannian submersion.

See also

Notes

References

  • Brezinski, Claude: Projection Methods for Systems of Equations, North-Holland, ISBN 0-444-82777-3 (1997).