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Concept
Projection (linear algebra)
Summary
In linear algebra and functional analysis, a projection is a linear transformation from a vector space to itself (an endomorphism) such that . That is, whenever is applied twice to any vector, it gives the same result as if it were applied once (i.e. is idempotent). It leaves its unchanged. 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.
A projection on a vector space is a linear operator such that .
When has an inner product and is complete (i.e. when is a Hilbert space) the concept of orthogonality can be used. A projection on a Hilbert space is called an orthogonal projection if it satisfies for all . A projection on a Hilbert space that is not orthogonal is called an oblique projection.
A square matrix is called a projection matrix if it is equal to its square, i.e. if .
A square matrix is called an orthogonal projection matrix if for a real matrix, and respectively for a complex matrix, where denotes the transpose of and denotes the adjoint or Hermitian transpose of .
A projection matrix that is not an orthogonal projection matrix is called an oblique projection matrix.
The eigenvalues of a projection matrix must be 0 or 1.
For example, the function which maps the point in three-dimensional space to the point 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 is indeed a projection, i.e., , we compute
Observing that shows that the projection is an orthogonal projection.
A simple example of a non-orthogonal (oblique) projection is
Via matrix multiplication, one sees that
showing that is indeed a projection.
The projection is orthogonal if and only if because only then
By definition, a projection is idempotent (i.e. ).
Every projection is an open map, meaning that it maps each open set in the domain to an open set in the subspace topology of the .
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