Template:Unite

See also projection (linear algebra).

In mathematics, a projection operator P on a vector space is a linear transformation that is idempotent, that is, P² = P. Such transformations project any vector in the vector space to a vector in the subspace that is the image of the transformation. In linear algebra, a projection is a linear transformation P such that P2 = P, i. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ...

In an inner product space, such an operator is an orthogonal projection if and only if it is self-adjoint. In finite-dimensional inner product spaces, an orthogonal projection matrix is a matrix P which satisfies P² = P and P^* = P where P^* is the conjugate transpose (adjoint operator) of P (see projection (linear algebra)). The condition that P^* = P says P is a symmetric matrix if all of the entries in P are real. In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In geometry, an orthogonal projection of a k-dimensional object onto a d-dimensional hyperplane (d < k) is obtained by intersections of (k &#8722; d)- dimensional hyperplanes drawn through the points of an object orthogonally to the d-hyperplane. ... In mathematics, an element x of a star-algebra is self-adjoint if the involution acts trivially upon it. ... In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... In linear algebra, a projection is a linear transformation P such that P2 = P, i. ... In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ...

In physics, the term projection operator usually means self-adjoint projection operator. The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ...

The only possible eigenvalues of a projection operator over a finite-dimensional real or complex vector space are 0 and 1. In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

Intuitively, a projection operator "picks out" entries in a vector, for example,