Projection (linear algebra)

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This article is about the use of "projection" in linear algebra. For other uses, see Projection.

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

In linear algebra, 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.

1 Simple example
2 Properties
3 Orthogonal projections
4 Oblique projections
5 Uses
6 See also
7 Notes
8 References

[edit] 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 $P = P 2$ , proving that P is indeed a projection.

[edit] Properties

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

Every projection P is an idempotent transformation, meaning that P² = P. It is also a linear transformation. These facts have many implications. 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, and the domain 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]

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.

[edit] Orthogonal projections

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 the associated matrix is symmetric: P = P^T (for complex-valued projections, the matrix must be 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

$P_u = u u^\top. \,$

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, and let U denote the n-by-k matrix whose columns are u₁, …, u_k. Then the projection is given by

$P_U = U U^\top. \,$ ^[5]

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

$P_U = U (U^\top U)^{-1} U^\top.$ ^[6]

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

[edit] 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 U. 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 V. Then the projection is defined by