Projection operator

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. 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.

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 x–y 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

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.

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:

The range and kernel of a projection are complementary, as are ${\displaystyle P}$ and ${\displaystyle Q=I-P}$. The operator ${\displaystyle Q}$ is also a projection and the range and kernel of ${\displaystyle P}$ become the kernel and range of ${\displaystyle Q}$ and vice versa. We say ${\displaystyle P}$ is a projection along V onto U (kernel/range) and ${\displaystyle Q}$ is a projection along U onto V.

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