Projection (linear algebra)

In infinite-dimensional vector spaces, the spectrum of a projection is contained in

In general, the corresponding eigenspaces are (respectively) the kernel and range of the projection.

Decomposition of a vector space into direct sums is not unique.

has an inner product and is complete (is a Hilbert space) the concept of orthogonality can be used.

For finite-dimensional complex or real vector spaces, the standard inner product can be substituted for

A simple case occurs when the orthogonal projection is onto a line.

is a unit vector on the line, then the projection is given by the outer product

This operator leaves u invariant, and it annihilates all vectors orthogonal to

[4] A simple way to see this is to consider an arbitrary vector

as the sum of a component on the line (i.e. the projected vector we seek) and another perpendicular to it,

by the properties of the dot product of parallel and perpendicular vectors.

This formula can be generalized to orthogonal projections on a subspace of arbitrary dimension.

into the underlying vector space but is no longer an isometry in general.

In the general case, we can have an arbitrary positive definite matrix

When the range space of the projection is generated by a frame (i.e. the number of generators is greater than its dimension), the formula for the projection takes the form:

Further details on sums of projectors can be found in Banerjee and Roy (2014).

[9] Also see Banerjee (2004)[10] for application of sums of projectors in basic spherical trigonometry.

Whereas calculating the fitted value of an ordinary least squares regression requires an orthogonal projection, calculating the fitted value of an instrumental variables regression requires an oblique projection.

form a basis for the orthogonal complement of the kernel of the projection, and assemble these vectors in the matrix

In general, if the vector space is over complex number field, one then uses the Hermitian transpose

over a field is a diagonalizable matrix, since its minimal polynomial divides

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[14] where

Many of the algebraic results discussed above survive the passage to this context.

into complementary subspaces still specifies a projection, and vice versa.

However, in contrast to the finite-dimensional case, projections need not be continuous in general.

Projections (orthogonal and otherwise) play a major role in algorithms for certain linear algebra problems: As stated above, projections are a special case of idempotents.

Analytically, orthogonal projections are non-commutative generalizations of characteristic functions.

Therefore, as one can imagine, projections are very often encountered in the context of operator algebras.

In particular, a von Neumann algebra is generated by its complete lattice of projections.

one can analogously ask for this map to be an isometry on the orthogonal complement of the kernel: that