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Projection (linear algebra)

  (Redirected from Orthogonal projection)
The transformation P is the orthogonal projection onto the line m.

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P 2 = P. That is, whenever P is applied twice to any value, it gives the same result as if it were applied once (idempotent). It leaves its image unchanged.[1] Though abstract, 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.


Simple exampleEdit

Orthogonal projectionEdit

For example, the function which maps the point (x, y, z) in three-dimensional space R3 to the point (x, y, 0) 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 P is indeed a projection, i.e., P = P2, we compute


Finding projection with an inner productEdit

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


where the i 's imply Einstein sum notation. y can be written as an orthogonal sum such that   + z.   is sometimes denoted as  . There is a theorem in Linear Algebra that states that this z is the shortest distance from y to V and is commonly used in areas such as machine learning.

y is being projected onto the vector space V.

Oblique projectionEdit

A simple example of a non-orthogonal (oblique) projection (for definition see below) is


Via matrix multiplication, one sees that


proving that P is indeed a projection.

The projection P is orthogonal if and only if α = 0.

Properties and classificationEdit

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

Let W be a finite dimensional vector space and P be a projection on W. Suppose the subspaces U and V are the range and kernel of P respectively. Then P has the following properties:

  1. By definition,   is idempotent (i.e.  ).
  2.   is the identity operator   on  
  3. We have a direct sum  . Every vector   may be decomposed uniquely as   with   and  , and where  .

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

In infinite dimensional vector spaces, the spectrum of a projection is contained in {0, 1} as


Only 0 or 1 can be an eigenvalue of a projection, implying that   is always a positive semi-definite matrix. The corresponding eigenspaces are (respectively) the kernel and range of the projection. Decomposition of a vector space into direct sums is not unique in general. 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 roots, and thus   is diagonalizable.

The product of projections is not, in general, a projection, even if they are orthogonal. If projections commute, then their product is a projection.

Orthogonal projectionsEdit

When the vector space W 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 U and the null space V are orthogonal subspaces. Thus, for every x and y in W,  . Equivalently:


A projection is orthogonal if and only if it is self-adjoint. Using the self-adjoint and idempotent properties of P, for any x and y in W we have PxU, yPyV, and


where   is the inner product associated with W. Therefore, Px and yPy are orthogonal.[2] The other direction, namely that if P is orthogonal then it is self-adjoint, follows from


for every x and y in W; thus P = P*.

Properties and special casesEdit

An orthogonal projection is a bounded operator. This is because for every v in the vector space we have, by Cauchy–Schwarz inequality:


Thus  .

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


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


(If u 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 u, proving that it is indeed the orthogonal projection onto the line containing u.[3] 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 u1, ..., uk be an orthonormal basis of the subspace U, and let A denote the n-by-k matrix whose columns are u1, ..., uk. Then the projection is given by:[4]


which can be rewritten as


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

The orthonormality condition can also be dropped. If u1, ..., uk is a (not necessarily orthonormal) basis, and A is the matrix with these vectors as columns, then the projection is:[5][6]


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

In the general case, we can have an arbitrary positive definite matrix D 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., B is the null space matrix of A),[7] the following holds:


If the orthogonal condition is enhanced to AT W B = AT WT B = 0 with W 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).[8] Also see Banerjee (2004)[9] for application of sums of projectors in basic spherical trigonometry.

Oblique projectionsEdit

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.

Projections are defined by their null space and the basis vectors used to characterize their range (which is the complement of the null space). When these basis vectors are orthogonal to the null space, then the projection is an orthogonal projection. When these basis vectors are not orthogonal to the null space, the projection is an oblique projection. Let the vectors u1, ..., uk form a basis for the range of the projection, and assemble these vectors in the n-by-k matrix A. The range and the null space are complementary spaces, so the null space has dimension nk. It follows that the orthogonal complement of the null space has dimension k. Let v1, ..., vk form a basis for the orthogonal complement of the null space of the projection, and assemble these vectors in the matrix B. Then the projection is defined by


This expression generalizes the formula for orthogonal projections given above.[10][11]

Canonical formsEdit

Any projection P = P2 on a vector space of dimension d over a field is a diagonalizable matrix, since its minimal polynomial divides x2x, which splits into distinct linear factors. Thus there exists a basis in which P has the form


where r is the rank of P. Here Ir is the identity matrix of size r, and 0dr is the zero matrix of size dr. 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[12]


where σ1σ2 ≥ ... ≥ σk > 0. The integers k, s, m and the real numbers   are uniquely determined. Note that 2k + s + m = d. The factor Im ⊕ 0s corresponds to the maximal invariant subspace on which P acts as an orthogonal projection (so that P itself is orthogonal if and only if k = 0) and the σi-blocks correspond to the oblique components.

Projections on normed vector spacesEdit

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 notions 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 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 U is a closed subspace of X. If there exists a closed subspace V such that X = UV, then the projection P with range U and kernel V is continuous. This follows from the closed graph theorem. Suppose xnx and Pxny. One needs to show that Px = y. Since U is closed and {Pxn} ⊂ U, y lies in U, i.e. Py = y. Also, xnPxn = (IP)xnxy. Because V is closed and {(IP)xn} ⊂ V, we have xyV, i.e. P(xy) = PxPy = Pxy = 0, which proves the claim.

The above argument makes use of the assumption that both U and V are closed. In general, given a closed subspace U, there need not exist a complementary closed subspace V, 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 U be the linear span of u. By Hahn–Banach, there exists a bounded linear functional φ such that φ(u) = 1. The operator P(x) = φ(x)u satisfies P2 = P, i.e. it is a projection. Boundedness of φ implies continuity of P and therefore ker(P) = ran(IP) is a closed complementary subspace of U.

Applications and further considerationsEdit

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 operator algebras. In particular, a von Neumann algebra is generated by its complete lattice of projections.


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 alsoEdit


  1. ^ Meyer, pp 386+387
  2. ^ Meyer, p. 433
  3. ^ Meyer, p. 431
  4. ^ Meyer, equation (5.13.4)
  5. ^ 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
  6. ^ Meyer, equation (5.13.3)
  7. ^ See also Linear least squares (mathematics) § Properties of the least-squares estimators.
  8. ^ 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
  9. ^ Banerjee, Sudipto (2004), "Revisiting Spherical Trigonometry with Orthogonal Projectors", The College Mathematics Journal, 35: 375–381, doi:10.1080/07468342.2004.11922099
  10. ^ 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
  11. ^ Meyer, equation (7.10.39)
  12. ^ Doković, D. Ž. (August 1991). "Unitary similarity of projectors". Aequationes Mathematicae. 42 (1): 220–224. doi:10.1007/BF01818492.


  • 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
  • Dunford, N.; Schwartz, J. T. (1958). Linear Operators, Part I: General Theory. Interscience.
  • Meyer, Carl D. (2000). Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Mathematics. ISBN 978-0-89871-454-8.

External linksEdit