Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P² = P. Projections map the whole vector space to a subspace and leave the points in that subspace 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 example

For example, the function which maps the point (x, y, z) in three-dimensional space to the point (x, y, 0) is a projection onto the x-y plane. This function is represented by the matrix

P={\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}}.

Indeed, the action of this matrix on an arbitrary vector is

P{\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}x\\y\\0\end{pmatrix}}

and

P^{2}{\begin{pmatrix}x\\y\\z\end{pmatrix}}=P{\begin{pmatrix}x\\y\\0\end{pmatrix}}={\begin{pmatrix}x\\y\\0\end{pmatrix}};

therefore P = P², proving that P is indeed a projection.

Classification

Assume the underlying vector space is finite dimensional (therefore issues such as continuity of a projection need not be considered).

As stated in the introduction, a projection P is a linear transformation that is idempotent, meaning that P² = P. Thus it has minimal polynomial $X^{2}-X=X(X-1)$ , which factors into distinct roots, and thus P is diagonalizable.

This can be seen more directly as follows: the image and kernel are the eigenspaces corresponding to 1 and 0, and give a direct sum decomposition of the space.

More explicitly: first, there is a subspace U of the domain for which the projection acts as the identity; every vector x in this subspace has Px = x. This subspace is exactly the range of the projection.

There is a complementary subspace V of the domain that is always zeroed out by the projection; every vector x in this subspace has Px = 0. This subspace is the null space of the projection.

The projection is said to be along V onto U. The subspaces U and V determine the projection uniquely.

The subspaces U and V are complementary, i.e. the underlying vector space is the direct sum U ⊕ V. This means that any vector x in the domain can uniquely be written as x = u + v with u in U and v in V. The vector u in this decomposition is given by u = Px, where P is the projection along V onto U. The vector v is given by v = (I − P) x. The operator I − P is the projection along U onto V; it is called the complementary projection.^[2] 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.

Only 0 and 1 can be an eigenvalue of a projection. The eigenspace corresponding to the eigenvalue 0 is the null space V, and the eigenspace corresponding to 1 is the range U.

Orthogonal projections

If the underlying vector space is endowed with an inner product, orthogonality and its attendant notions (such as the self-adjointness of a linear operator) become available. An orthogonal projection is a projection for which the range U and the null space V are orthogonal subspaces. A projection is orthogonal if and only if it is self-adjoint, which means that, in the context of real vector spaces, the associated matrix is symmetric relative to an orthonormal basis: P = P^T (for the complex case, the matrix is hermitian: P = P^*). Indeed, if x is a vector in the domain of the projection, then Px ∈ U and x − Px ∈ V, and

(Px)^{\top }(x-Px)=x^{\top }(P-P^{2})x,\,

so Px and x − Px are orthogonal for all x if and only if P − P² = 0.^[3]

The simplest case is where the projection is an orthogonal projection onto a line. If u is a unit vector on the line, then the projection is given by

P_{u}=uu^{\top }.\,

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.^[4]

This formula can be generalized to orthogonal projections on a subspace of arbitrary dimension. Let u₁, …, u_k be an orthonormal basis of the subspace U, and let A denote the n-by-k matrix whose columns are u₁, …, u_k. Then the projection is given by

P_{A}=AA^{\top }.\,

^[5]

The matrix A^T is the partial isometry that vanishes on the orthogonal complement of U and A is the isometry that embedds U into the underlying vector space. The range of P_A is therefore the final space of A. It is also clear that A^TA is the identity operator on U.

The orthonormality condition can also be dropped. If u₁, …, u_k is a (not necessarily orthonormal) basis, and A is the matrix with these vectors as columns, then the projection is

P_{A}=A(A^{\top }A)^{-1}A^{\top }.

^[6]

The matrix A^T still embeds U into the underlying vector space but is no longer an isometry in general. The matrix (A^TA)⁻¹ is a "normalizing factor" that recovers the norm. For example, the rank-1 operator uu^T is not a projection if ||u|| ≠ 1. After dividing by u^Tu = ||u||², we obtain the projection u(u^Tu)⁻¹u^T onto the subspace spanned by u.

All these formulas also hold for complex inner product spaces, provided that the conjugate transpose is used instead of the transpose.

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.

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 u₁, …, u_k 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 v₁, …, v_k 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

P=A(B^{\top }A)^{-1}B^{\top }.

This expression generalizes the formula for orthogonal projections given above.^[7]

Projections on normed vector spaces

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.

Much 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 P² = P. Conversely, if P is projection on X, i.e. P² = P, then it is easily verified that (I − P)² = (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).

However, in contrast to the finite-dimensional case, projections need not be 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) = Ran(P) ⊕ Ran(I − P).

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_n → x and Px_n → y. One needs to show Px = y. Since U is closed and {Px_n} ⊂ U, y lies in U, i.e. Py = y. Also, x_n − Px_n = (I − P)x_n → x − y. Because V is closed and {(I − P)x_n} ⊂ V, we have x − y ∈ V, i.e. P(x − y) = Px − Py = Px − y = 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(I − P) is a closed complementary subspace of U.

Applications and further considerations

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

QR decomposition (see Householder transformation and Gram-Schmidt decomposition);
Singular value decomposition
Reduction to Hessenberg form (the first step in many eigenvalue algorithms).

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.

Notes

^ Meyer, pp 386+387
^ Meyer, pp 383–388
^ Meyer, p. 433
^ Meyer, p. 431
^ Meyer, equation (5.13.4)
^ Meyer, equation (5.13.3)
^ Meyer, equation (7.10.39)

References

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

External Links

MIT Linear Algebra Lecture on Projection Matrices by GlobalVersity at Google Video, from MIT OpenCourseWare

[1] Meyer, pp 386+387

[2] Meyer, pp 383–388

[3] Meyer, p. 433

[4] Meyer, p. 431

[5] Meyer, equation (5.13.4)

[6] Meyer, equation (5.13.3)

[7] Meyer, equation (7.10.39)

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