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| {{redirect6|Orthogonal projection|the technical drawing concept|Orthographic projection|a concrete discussion of orthogonal projections in finite-dimensional linear spaces |Vector projection}}
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| [[File:Orthogonal projection.svg|frame|right|The transformation ''P'' is the orthogonal projection onto the line ''m''.]]
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| In [[linear algebra]] and [[functional analysis]], a '''projection''' is a [[linear transformation]] ''P'' from a [[vector space]] to itself such that {{nowrap|1=''P''<sup>2</sup> = ''P''}}. That is, whenever ''P'' is applied twice to any value, it gives the same result as if it were applied once ([[Idempotence|idempotent]]). It leaves its image unchanged.<ref>Meyer, pp 386+387</ref>
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| 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 [[geometry|geometrical]] object by examining the effect of the projection on [[point (geometry)|point]]s in the object.
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| == Simple example ==
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| ===Orthogonal projection===
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| For example, the function which maps the point (''x'', ''y'', ''z'') in three-dimensional space '''R'''<sup>3</sup> to the point (''x'', ''y'', 0) is a projection onto the ''x''–''y'' plane. This function is represented by the [[matrix (mathematics)|matrix]]
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| :<math> P = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}. </math>
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| The action of this matrix on an arbitrary vector is
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| :<math> P \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix}
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| x \\ y \\ 0 \end{pmatrix}.</math>
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| To see that ''P'' is indeed a projection, i.e., ''P'' = ''P''<sup>2</sup>, we compute: | |
| <math> P^2 \begin{pmatrix} x \\ y \\ z \end{pmatrix} = P \begin{pmatrix} x \\ y \\ 0 \end{pmatrix} = \begin{pmatrix} x \\ y \\ 0 \end{pmatrix} = P\begin{pmatrix} x \\ y \\ z \end{pmatrix}. </math>
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| ===Oblique projection===
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| A simple example of a non-orthogonal (oblique) projection (for definition see below) is
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| :<math> P = \begin{bmatrix} 0 & 0 \\ \alpha & 1 \end{bmatrix}. </math>
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| Via [[matrix multiplication]], one sees that
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| :<math> P^2 = \begin{bmatrix} 0 & 0 \\ \alpha & 1 \end{bmatrix} \begin{bmatrix} 0 & 0 \\ \alpha & 1 \end{bmatrix}
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| = \begin{bmatrix} 0 & 0 \\ \alpha & 1 \end{bmatrix} = P. </math>
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| proving that ''P'' is indeed a projection.
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| The projection P is orthogonal if and only if <math> \alpha </math> = 0.
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| == Properties and classification ==
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| [[File:Oblique projection.svg|frame|right|The transformation ''T'' is the projection along ''k'' onto ''m''. The range of ''T'' is ''m'' and the null space is ''k''.]]
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| Let ''W'' be a finite dimensional vector space. Suppose the [[Linear subspace|subspace]]s ''U'' and ''V'' are the [[range of a matrix|range]] and [[null space|kernel]] of ''P'' respectively.
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| Then ''P'' has the following basic properties:
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| # ''P'' is the identity operator ''I'' on ''U'': <math>\forall x \in U: Px = x.</math>
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| # We have a [[direct sum of vector spaces|direct sum]] ''W'' = ''U'' ⊕ ''V''. Every vector ''x'' in ''W'' may be decomposed uniquely as ''x'' = ''u'' + ''v'' with <math>u = Px</math> and <math>v = x - Px = (I - P)x</math>, and where ''u'' is in ''U'' and ''v'' is in ''V''.
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| # ''P'' is [[idempotent]] and satisfies ''P''<sup>2</sup> = ''P''.
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| The range and kernel of a projection are ''complementary'', as are ''P'' and ''Q'' = ''I'' − ''P''. The operator ''Q'' is also a projection and the range and kernel of P become the kernel and range of Q and vice-versa. We say ''P'' is a projection along ''V'' onto ''U'' (kernel/range) and ''Q'' is a projection along ''U'' onto ''V''.
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| In infinite dimensional vector spaces
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| [[Spectrum (functional analysis)|spectrum]] of a projection is contained in {0, 1}, as <math> (\lambda I - P)^{-1}= \frac 1 \lambda I+\frac 1{\lambda(\lambda-1)} P</math>. Only 0 and 1 can be an [[eigenvalue]] of a projection. The corresponding eigenspaces are (respectively) the kernel and range of the projection.
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| Decomposition of a vector space into direct sums is not unique in general. Therefore, given a subspace ''V'', in general there are many projections whose range (or kernel) is ''V''.
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| If a projection is nontrivial it has [[minimal polynomial (linear algebra)|minimal polynomial]] <math>X^2-X=X(X- I)</math>, which factors into distinct roots, and thus ''P'' is [[diagonalizable]].
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| ===Orthogonal projections===
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| When the vector space ''W'' has an [[inner product]] (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 [[orthogonality|orthogonal subspaces]]. A projection is orthogonal if and only if it is [[self-adjoint operator|self-adjoint]]. Using the self-adjoint and idempotent properties of ''P'', for any ''x'' and ''y'' in ''W'' we have ''Px'' ∈ ''U'', ''y'' − ''Py'' ∈ ''V'', and
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| :<math> \langle Px, y-Py \rangle = \langle P^2x, y-Py \rangle = \langle Px, P(I-P)y \rangle = \langle Px, (P-P^2)y \rangle = 0 \,</math>
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| where <math>\langle\cdot,\cdot\rangle</math> is the [[inner product]] associated with ''W''. Therefore, ''Px'' and ''y'' − ''Py'' are orthogonal.<ref>Meyer, p. 433</ref>
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| For finite dimensional complex or real vector spaces, the [[standard inner product]] can be substituted for <math>\langle\cdot,\cdot\rangle</math>.
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| 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
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| :<math> P_u = u u^\mathrm{T}. \, </math>
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| 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''.<ref>Meyer, p. 431</ref> A simple way to see this is to consider an arbitrary vector <math>x</math> as the sum of a component on the line (i.e. the projected vector we seek) and another perpendicular to it, <math>x=x_\parallel+x_\perp</math>. Applying projection, we get <math>P_ux=u u^\mathrm{T} x_\parallel+u u^\mathrm{T} x_\perp=u|x_\parallel|+u0=x_\parallel</math> by the properties of the [[dot product]] of parallel and perpendicular vectors.
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| This formula can be generalized to orthogonal projections on a subspace of arbitrary dimension. Let ''u''<sub>1</sub>, ..., ''u''<sub>''k''</sub> be an [[orthonormal basis]] of the subspace ''U'', and let ''A'' denote the ''n''-by-''k'' matrix whose columns are ''u''<sub>1</sub>, ..., ''u''<sub>''k''</sub>. Then the projection is given by
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| :<math> P_A = A A^\mathrm{T} \, </math><ref>Meyer, equation (5.13.4)</ref>
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| which can be rewritten as
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| <math> P_A = \sum_i \langle u_i,\cdot\rangle u_i.</math>
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| The matrix ''A''<sup>T</sup> 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 ''P<sub>A</sub>'' is therefore the ''final space'' of ''A''. It is also clear that ''A''<sup>T</sup>''A'' is the identity operator on ''U''.
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| The orthonormality condition can also be dropped. If ''u''<sub>1</sub>, ..., ''u''<sub>''k''</sub> is a (not necessarily orthonormal) basis, and ''A'' is the matrix with these vectors as columns, then the projection is
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| :<math>P_A = A (A^\mathrm{T} A)^{-1} A^\mathrm{T}. </math><ref>Meyer, equation (5.13.3)</ref>
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| The matrix ''A'' still embeds ''U'' into the underlying vector space but is no longer an isometry in general. The matrix (''A''<sup>T</sup>''A'')<sup>−1</sup> is a "normalizing factor" that recovers the norm. For example, the rank-1 operator ''uu''<sup>T</sup> is not a projection if ||''u''|| ≠ 1. After dividing by ''u''<sup>T</sup>''u'' = ||''u''||<sup>2</sup>, we obtain the projection ''u''(''u''<sup>T</sup>''u'')<sup>−1</sup>''u''<sup>T</sup> onto the subspace spanned by ''u''.
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| When the range space of the projection is generated by a [[Frame of a vector space|frame]] (i.e. the number of generators is greater than its dimension), the formula for the projection takes the form
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| <math>P_A = A (A^\mathrm{T} A)^+ A^\mathrm{T}</math>. Here <math>A^+</math> stands for the [[Moore–Penrose pseudoinverse]]. This is just one of many ways to construct the projection operator.
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| If a matrix <math>\, [A \ B]</math> is non-singular and ''A''<sup>T</sup> ''B'' = 0 (i.e., ''B'' is the [[null space]] matrix of ''A''),<ref>see also [[Linear_least_squares_(mathematics)#Properties_of_the_least-squares_estimators|Properties of the least-squares estimators in Linear least squares]]</ref> the following holds:
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| :<math>I = [A \ B][A \ B]^{-1} \begin{bmatrix}A^\mathrm{T}\\B^\mathrm{T}\end{bmatrix}^{-1}\begin{bmatrix}A^\mathrm{T}\\B^\mathrm{T}\end{bmatrix} = [A \ B](\begin{bmatrix}A^\mathrm{T}\\B^\mathrm{T}\end{bmatrix}[A \ B])^{-1} \begin{bmatrix}A^\mathrm{T}\\B^\mathrm{T}\end{bmatrix}
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| = [A \ B] \begin{bmatrix}A^\mathrm{T}A & O \\ O & B^\mathrm{T}B\end{bmatrix}^{-1} \begin{bmatrix}A^\mathrm{T}\\B^\mathrm{T}\end{bmatrix}</math>
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| :<math>= A (A^\mathrm{T} A)^{-1} A^\mathrm{T} + B (B^\mathrm{T} B)^{-1} B^\mathrm{T}. </math>
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| If the orthogonal condition is enhanced to ''A''<sup>T</sup> ''W'' ''B'' = ''A''<sup>T</sup> ''W''<sup>T</sup> ''B'' = 0 with ''W'' being non-singular<!-- and symmetric-->, the following holds:
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| :<math>I = \begin{bmatrix}A & B\end{bmatrix} \begin{bmatrix}(A^\mathrm{T} W A)^{-1} A^\mathrm{T} \\ (B^\mathrm{T} W B)^{-1} B^\mathrm{T} \end{bmatrix} W .</math>
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| All these formulas also hold for complex inner product spaces, provided that the [[conjugate transpose]] is used instead of the transpose.
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| === Oblique projections ===
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| 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.
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| Oblique projections are defined by their range and null space. A formula for the matrix representing the projection with a given range and null space can be found as follows. Let the vectors ''u''<sub>1</sub>, ..., ''u''<sub>''k''</sub> 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 ''n'' − ''k''. It follows that the [[orthogonal complement]] of the null space has dimension ''k''. Let ''v''<sub>1</sub>, ..., ''v''<sub>''k''</sub> 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
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| :<math> P = A (B^\mathrm{T} A)^{-1} B^\mathrm{T}. </math>
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| This expression generalizes the formula for orthogonal projections given above.<ref>Meyer, equation (7.10.39)</ref>
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| == Canonical forms ==
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| Any projection {{nowrap|''P'' {{=}} ''P''<sup>2</sup>}} on a vector space of dimension ''d'' over a field is a [[diagonalizable matrix]], since its [[minimal polynomial (linear algebra)|minimal polynomial]] is ''x''<sup>2</sup> − ''x'', which splits into distinct linear factors. Thus there exists a basis in which ''P'' has the form
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| :<math>P = I_r\oplus 0_{d-r}</math>
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| where ''r'' is the rank of ''P''. Here ''I''<sub>''r''</sub> is the identity matrix of size ''r'', and 0<sub>''d''−''r''</sub> is the zero matrix of size ''d'' − ''r''. 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<ref>{{ cite journal|last=Doković|first= D. Ž. |title=Unitary similarity of projectors|journal=Aequationes Mathematicae| volume =42| issue= 1| pages= 220–224|date=August 1991|url=http://www.springerlink.com/content/w3r57501226447m6/|doi=10.1007/BF01818492}}</ref>
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| :<math>P = \begin{bmatrix}1&\sigma_1 \\ 0&0\end{bmatrix} \oplus \cdots \oplus \begin{bmatrix}1&\sigma_k \\ 0&0\end{bmatrix} \oplus I_m \oplus 0_s</math> .
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| where {{nowrap|σ<sub>1</sub> ≥ σ<sub>2</sub> ≥ ... ≥ σ<sub>''k''</sub> > 0}}. The integers ''k'', ''s'', ''m'' and the real numbers <math>\sigma_i</math> are uniquely determined. Note that {{nowrap|2''k'' + ''s'' + ''m'' {{=}} ''d''}}. The factor {{nowrap|''I''<sub>''m''</sub> ⊕ 0<sub>''s''</sub>}} 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 σ<sub>''i''</sub>-blocks correspond to the ''oblique'' components.
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| == Projections on normed vector spaces == | |
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| When the underlying vector space ''X'' is a (not necessarily finite-dimensional) [[normed vector space]], analytic questions, irrelevant in the finite-dimensional case, need to be considered. Assume now ''X'' is a [[Banach space]].
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| Many of the algebraic notions discussed above survive the passage to this context. A given direct sum decomposition of ''X'' into complementary subspaces still specifies a projection, and vice versa. If ''X'' is the direct sum ''X'' = ''U'' ⊕ ''V'', then the operator defined by ''P''(''u'' + ''v'') = ''u'' is still a projection with range ''U'' and kernel ''V''. It is also clear that ''P''<sup>2</sup> = ''P''. Conversely, if ''P'' is projection on ''X'', i.e. ''P''<sup>2</sup> = ''P'', then it is easily verified that (''I'' − ''P'')<sup>2</sup> = (''I'' − ''P''). In other words, (''I'' − ''P'') is also a projection. The relation ''I'' = ''P'' + (''I'' − ''P'') implies ''X'' is the direct sum Ran(''P'') ⊕ Ran(''I'' − ''P'').
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| However, in contrast to the finite-dimensional case, projections need not be [[bounded linear operator|continuous]] in general. If a subspace ''U'' of ''X'' is not closed in the norm topology, then projection onto ''U'' is not continuous. In other words, the range of a continuous projection ''P'' 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 ''P'' gives a decomposition of ''X'' into two complementary ''closed'' subspaces: ''X'' = Ran(''P'') ⊕ Ker(''P'') = Ker(''I'' − ''P'') ⊕ Ker(''P'').
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| 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'' = ''U'' ⊕ ''V'', then the projection ''P'' with range ''U'' and kernel ''V'' is continuous. This follows from the [[closed graph theorem]]. Suppose ''x<sub>n</sub>'' → ''x'' and ''Px<sub>n</sub>'' → ''y''. One needs to show ''Px'' = ''y''. Since ''U'' is closed and {''Px<sub>n</sub>''} ⊂ ''U'', ''y'' lies in ''U'', i.e. ''Py'' = ''y''. Also, ''x<sub>n</sub>'' − ''Px<sub>n</sub>'' = (''I'' − ''P'')''x<sub>n</sub>'' → ''x'' − ''y''. Because ''V'' is closed and {(''I'' − ''P'')''x<sub>n</sub>''} ⊂ ''V'', we have ''x'' − ''y'' ∈ ''V'', i.e. ''P''(''x'' − ''y'') = ''Px'' − ''Py'' = ''Px'' − ''y'' = 0, which proves the claim.
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| 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 space]]s 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 ''P''<sup>2</sup> = ''P'', i.e. it is a projection. Boundedness of ''φ'' implies continuity of ''P'' and therefore Ker(''P'') = Ran(''I'' − ''P'') is a closed complementary subspace of ''U''.
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| However, every continuous projection on a Banach space is an [[open mapping]], by the [[Open mapping theorem (functional analysis)|open mapping theorem]].
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| == Applications and further considerations ==
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| Projections (orthogonal and otherwise) play a major role in [[algorithm]]s for certain linear algebra problems:
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| * [[QR decomposition]] (see [[Householder transformation]] and [[Gram–Schmidt decomposition]]);
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| * [[Singular value decomposition]]
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| * Reduction to [[Hessenberg matrix|Hessenberg]] form (the first step in many [[eigenvalue algorithm]]s).
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| * [[Linear regression]]
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| As stated above, projections are a special case of idempotents. Analytically, orthogonal projections are non-commutative generalizations of [[characteristic polynomial|characteristic functions]]. Idempotents are used in classifying, for instance, [[semisimple algebra]]s, 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 algebra]]s. In particular, a [[von Neumann algebra]] is generated by its complete [[lattice (order)|lattice]] of projections.
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| == Generalizations ==
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| More generally, given a map between normed vector spaces <math>T\colon V \to W,</math> one can analogously ask for this map to be an isometry on the orthogonal complement of the kernel: that <math>(\ker T)^\perp \to W</math> be an 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]].
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| ==See also==
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| *[[Centering matrix]], which is an example of a projection matrix.
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| *[[Orthogonalization]]
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| *[[Invariant subspace]]
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| *[[Trace (linear algebra)#Properties|Properties of trace]]
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| *[[Dykstra's projection algorithm]] to compute the projection onto an intersection of sets
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{cite book|first1=N. |last1=Dunford |first2= J. T.|title=Linear Operators, Part I: General Theory|publisher= Interscience|year= 1958|last2=Schwartz}}
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| * {{cite book|first=Carl D.|last= Meyer|url=http://www.matrixanalysis.com/ |title=Matrix Analysis and Applied Linear Algebra|publisher=Society for Industrial and Applied Mathematics|year= 2000| isbn= 978-0-89871-454-8}}
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| ==External links==
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| * [http://www.youtube.com/watch?v=osh80YCg_GM&feature=PlayList&p=38823D6325151CED&index=16 MIT Linear Algebra Lecture on Projection Matrices] at Google Video, from MIT OpenCourseWare
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| * [http://www.mtsu.edu/~csjudy/planeview3D/tutorial.html Planar Geometric Projections Tutorial] - a simple-to-follow tutorial explaining the different types of planar geometric projections.
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| {{linear algebra}}
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| [[Category:Functional analysis]]
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| [[Category:Linear algebra]]
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| [[Category:Linear operators]]
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| [[de:Projektion (Mathematik)]]
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| [[fi:Projektio (lineaarialgebra)]]
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