The transformation P is the orthogonal projection onto the line m.

In linear algebra and functional analysis, a projection is a linear transformation from a vector space to itself (an endomorphism) such that . That is, whenever is applied twice to any vector, it gives the same result as if it were applied once (i.e. is idempotent). It leaves its image unchanged.[1] 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 points in the object.

Definitions

A projection on a vector space is a linear operator such that .

When has an inner product and is complete, i.e. when is a Hilbert space, the concept of orthogonality can be used. A projection on a Hilbert space is called an orthogonal projection if it satisfies for all . 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

For example, the function which maps the point in three-dimensional space to the point is an orthogonal projection onto the xy-plane. This function is represented by the matrix

The action of this matrix on an arbitrary vector is

To see that is indeed a projection, i.e., , we compute

Observing that shows that the projection is an orthogonal projection.

Oblique projection

A simple example of a non-orthogonal (oblique) projection is

Via matrix multiplication, one sees that

showing that is indeed a projection.

The projection is orthogonal if and only if because only then

Properties and classification

The transformation T is the projection along k onto m. The range of T is m and the kernel is k.

Idempotence

By definition, a projection is idempotent (i.e. ).

Open map

Every projection is an open map, meaning that it maps each open set in the domain to an open set in the subspace topology of the image.[citation needed] That is, for any vector and any ball (with positive radius) centered on , there exists a ball (with positive radius) centered on that is wholly contained in the image .

Complementarity of image and kernel

Let be a finite-dimensional vector space and be a projection on . Suppose the subspaces and are the image and kernel of respectively. Then has the following properties:

  1. is the identity operator on :
  2. We have a direct sum . Every vector may be decomposed uniquely as with and , and where

The image and kernel of a projection are complementary, as are and . The operator is also a projection as the image and kernel of become the kernel and image of and vice versa. We say is a projection along onto (kernel/image) and is a projection along onto .

Spectrum

In infinite-dimensional vector spaces, the spectrum of a projection is contained in as

Only 0 or 1 can be an eigenvalue of a projection. This implies that an orthogonal projection is always a positive semi-definite matrix. In general, the corresponding eigenspaces are (respectively) the kernel and range of the projection. Decomposition of a vector space into direct sums is not unique. Therefore, given a subspace , there may be many projections whose range (or kernel) is .

If a projection is nontrivial it has minimal polynomial , which factors into distinct linear factors, and thus 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 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

Main articles: Hilbert projection theorem and Complemented subspace

When the vector space has an inner product and is complete (is a Hilbert space) the concept of orthogonality can be used. An orthogonal projection is a projection for which the range and the kernel are orthogonal subspaces. Thus, for every and in , . Equivalently:

A projection is orthogonal if and only if it is self-adjoint. Using the self-adjoint and idempotent properties of , for any and in we have , , and

where is the inner product associated with . Therefore, and are orthogonal projections.[3] The other direction, namely that if is orthogonal then it is self-adjoint, follows from the implication from to
for every and in ; thus .

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. This is because for every in the vector space we have, by the Cauchy–Schwarz inequality:

Thus .

For finite-dimensional complex or real vector spaces, the standard inner product can be substituted for .

Formulas

A simple case occurs when the orthogonal projection is onto a line. If is a unit vector on the line, then the projection is given by the outer product

(If is complex-valued, the transpose in the above equation is replaced by a Hermitian transpose). This operator leaves u invariant, and it annihilates all vectors orthogonal to , proving that it is indeed the orthogonal projection onto the line containing u.[4] A simple way to see this is to consider an arbitrary vector as the sum of a component on the line (i.e. the projected vector we seek) and another perpendicular to it, . Applying projection, we get
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. Let be an orthonormal basis of the subspace , with the assumption that the integer , and let denote the matrix whose columns are , i.e., . Then the projection is given by:[5]

which can be rewritten as

The matrix is the partial isometry that vanishes on the orthogonal complement of , and is the isometry that embeds into the underlying vector space. The range of is therefore the final space of . It is also clear that is the identity operator on .

The orthonormality condition can also be dropped. If is a (not necessarily orthonormal) basis with , and is the matrix with these vectors as columns, then the projection is:[6][7]

The matrix still embeds into the underlying vector space but is no longer an isometry in general. The matrix is a "normalizing factor" that recovers the norm. For example, the rank-1 operator is not a projection if After dividing by we obtain the projection onto the subspace spanned by .

In the general case, we can have an arbitrary positive definite matrix defining an inner product , and the projection is given by . Then

When the range space of the projection is generated by a frame (i.e. the number of generators is greater than its dimension), the formula for the projection takes the form: . Here stands for the Moore–Penrose pseudoinverse. This is just one of many ways to construct the projection operator.

If is a non-singular matrix and (i.e., is the null space matrix of ),[8] the following holds:

If the orthogonal condition is enhanced to with non-singular, the following holds:

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).[9] Also see Banerjee (2004)[10] 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

Let be a linear operator, such that and assume that is not the zero operator. Let the vectors form a basis for the range of , and assemble these vectors in the matrix . Therefore the integer , otherwise and is the zero operator. The range and the kernel are complementary spaces, so the kernel has dimension . It follows that the orthogonal complement of the kernel has dimension . Let form a basis for the orthogonal complement of the kernel of the projection, and assemble these vectors in the matrix . Then the projection (with the condition ) is given by

This expression generalizes the formula for orthogonal projections given above.[11][12] A standard proof of this expression is the following. For any vector in the vector space , we can decompose , where vector is in the image of , and vector So , and then is in the kernel of , which is the null space of In other words, the vector is in the column space of so for some dimension vector and the vector satisfies by the construction of . Put these conditions together, and we find a vector so that . Since matrices and are of full rank by their construction, the -matrix is invertible. So the equation gives the vector In this way, for any vector and hence .

In the case that is an orthogonal projection, we can take , and it follows that . By using this formula, one can easily check that . In general, if the vector space is over complex number field, one then uses the Hermitian transpose and has the formula . Recall that one can define the Moore–Penrose inverse of the matrix by since has full column rank, so .

Singular values

Note that is also an oblique projection. The singular values of and can be computed by an orthonormal basis of . Let be an orthonormal basis of and let be the orthogonal complement of . Denote the singular values of the matrix by the positive values . With this, the singular values for are:[13]

and the singular values for are
This implies that the largest singular values of and are equal, and thus that the matrix norm of the oblique projections are the same. However, the condition number satisfies the relation , and is therefore not necessarily equal.

Finding projection with an inner product

Let be a vector space (in this case a plane) spanned by orthogonal vectors . Let be a vector. One can define a projection of onto as

where repeated indices are summed over (Einstein sum notation). The vector can be written as an orthogonal sum such that . is sometimes denoted as . There is a theorem in linear algebra that states that this is the smallest distance (the orthogonal distance) from to and is commonly used in areas such as machine learning.

y is being projected onto the vector space V.

Canonical forms

Any projection on a vector space of dimension over a field is a diagonalizable matrix, since its minimal polynomial divides , which splits into distinct linear factors. Thus there exists a basis in which has the form

where is the rank of . Here is the identity matrix of size , is the zero matrix of size , and 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[14]

where . The integers and the real numbers are uniquely determined. Note that . The factor corresponds to the maximal invariant subspace on which acts as an orthogonal projection (so that P itself is orthogonal if and only if ) and the -blocks correspond to the oblique components.

Projections on normed vector spaces

When the underlying vector space is a (not necessarily finite-dimensional) normed vector space, analytic questions, irrelevant in the finite-dimensional case, need to be considered. Assume now is a Banach space.

Many of the algebraic results discussed above survive the passage to this context. A given direct sum decomposition of into complementary subspaces still specifies a projection, and vice versa. If is the direct sum , then the operator defined by is still a projection with range and kernel . It is also clear that . Conversely, if is projection on , i.e. , then it is easily verified that . In other words, is also a projection. The relation implies and is the direct sum .

However, in contrast to the finite-dimensional case, projections need not be continuous in general. If a subspace of is not closed in the norm topology, then the projection onto is not continuous. In other words, the range of a continuous projection 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 projection gives a decomposition of into two complementary closed subspaces: .

The converse holds also, with an additional assumption. Suppose is a closed subspace of . If there exists a closed subspace such that X = UV, then the projection with range and kernel is continuous. This follows from the closed graph theorem. Suppose xnx and Pxny. One needs to show that . Since is closed and {Pxn} ⊂ U, y lies in , i.e. Py = y. Also, xnPxn = (IP)xnxy. Because is closed and {(IP)xn} ⊂ V, we have , i.e. , which proves the claim.

The above argument makes use of the assumption that both and are closed. In general, given a closed subspace , there need not exist a complementary closed subspace , 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. Let be the linear span of . By Hahn–Banach, there exists a bounded linear functional such that φ(u) = 1. The operator satisfies , i.e. it is a projection. Boundedness of implies continuity of and therefore is a closed complementary subspace of .

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

More generally, given a map between normed vector spaces one can analogously ask for this map to be an isometry on the orthogonal complement of the kernel: that 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

  1. ^ Meyer, pp 386+387
  2. ^ a b Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis, second edition. Cambridge University Press. ISBN 9780521839402.
  3. ^ Meyer, p. 433
  4. ^ Meyer, p. 431
  5. ^ Meyer, equation (5.13.4)
  6. ^ Banerjee, Sudipto; Roy, Anindya (2014), Linear Algebra and Matrix Analysis for Statistics, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC, ISBN 978-1420095388
  7. ^ Meyer, equation (5.13.3)
  8. ^ See also Linear least squares (mathematics) § Properties of the least-squares estimators.
  9. ^ Banerjee, Sudipto; Roy, Anindya (2014), Linear Algebra and Matrix Analysis for Statistics, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC, ISBN 978-1420095388
  10. ^ Banerjee, Sudipto (2004), "Revisiting Spherical Trigonometry with Orthogonal Projectors", The College Mathematics Journal, 35 (5): 375–381, doi:10.1080/07468342.2004.11922099, S2CID 122277398
  11. ^ Banerjee, Sudipto; Roy, Anindya (2014), Linear Algebra and Matrix Analysis for Statistics, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC, ISBN 978-1420095388
  12. ^ Meyer, equation (7.10.39)
  13. ^ Brust, J. J.; Marcia, R. F.; Petra, C. G. (2020), "Computationally Efficient Decompositions of Oblique Projection Matrices", SIAM Journal on Matrix Analysis and Applications, 41 (2): 852–870, doi:10.1137/19M1288115, OSTI 1680061, S2CID 219921214
  14. ^ Doković, D. Ž. (August 1991). "Unitary similarity of projectors". Aequationes Mathematicae. 42 (1): 220–224. doi:10.1007/BF01818492. S2CID 122704926.

References