The concept of orthogonality is used within the mathematics in a number of different but related meanings. In elementary geometry is called two straight lines or planes orthogonal if they form a right angle, ie an angle of 90 °. In linear algebra, the concept is then extended to more general vector spaces and two vectors are called orthogonal if their dot product is zero. This meaning is then transferred to maps between vector spaces that allow the scalar and thus the orthogonality of two vectors unchanged.
- 3.1 Orthogonal and orthonormal vectors
- 3.2 Orthogonal Functions
- 3.3 Orthogonal Matrices
- 3.4 Orthogonal pictures
- 3.5 Orthogonal Projections
The term orthogonal (Greek ὀρθός orthos "right, right " and γωνία gonia " corner, angle" ) means " at right angles ". Equivalent to a right angle is also normal (Latin norma " measure " within the meaning of the right angle ). The term is normally used but much more comprehensive in mathematics. Perpendicular comes from the plumb line (Lot) and originally meant only orthogonal to the surface ( perpendicular ). The same fact is expressed by vertical (Latin vertex " apex ").
Refers to two straight lines, or vectors and planes that are orthogonal or non- orthogonal to each other, with
Based on the English term Perpendicular is the Orthogonalitätssymbol in HTML with ⊥ and in LaTeX perp coded ( within mathematics ) environment with \. The character encoding standard Unicode symbol ⊥ has the position U 27 C2.
Orthogonality in the geometry
In elementary geometry, two lines or planes are called orthogonal if they form a right angle, ie an angle of 90 °. The following designations are used:
- A straight line is called orthogonal ( normal ) to a level when her direction vector is a normal vector of the plane.
- One level is called orthogonal (normal level ) of a plane if their normal vector lies in this plane.
- A line / plane is orthogonal ( normal ) to a cam when it is perpendicular to the tangent / tangential plane in the intersection point.
In an orthogonal polygon (e.g. a rectangle) forming two adjacent sides of a right angle, wherein an orthogonal polyhedron (e.g. a cube ) two adjacent edges and therefore adjacent the side surfaces.
The angle between two vectors and in the Cartesian coordinate system can be on the dot product
Calculate. In this case, and in each case denote the lengths of the vectors, and the cosine of the included angle of the two vectors. Forming two vectors, and a right angle, the following applies
Two vectors thus are called orthogonal if their dot product is zero. The zero vector is orthogonal to all vectors. For example, two straight lines in the Euclidean plane with the slopes and then exactly orthogonal to each other if and only if.
In the synthetic geometry can orthogonality introduced by the axiomatic description of an orthogonality relation between straight lines on certain affine incidence levels.
Orthogonality in linear algebra
Orthogonal and orthonormal vectors
In linear algebra, Euclidean space are also included multi-dimensional vector spaces over the real or complex numbers in an extension of the term for which a scalar product is defined. The scalar product of two vectors and is a figure that must satisfy certain axioms and is typically written in the form. General then apply two vectors and a scalar product such as orthogonal to each other, if the dot product of two vectors is equal to zero, ie when
Applies. For example, in the space, the two vectors and with respect to the orthogonal Standardskalarprodukts as
Is. A set of vectors is then called orthogonal or orthogonal if all vectors contained in it are pairwise orthogonal. In addition, if all the vectors contained in it have the standard one, it is called the amount orthonormal or an orthonormal system. A set of orthogonal vectors that are all different from the zero vector is always linearly independent and therefore form a basis of the linear hull of this set. A basis of a vector space of orthonormal vectors is accordingly called orthonormal basis. Is paid for every two vectors an orthonormal basis
Wherein the Kronecker delta designated. Finite dimensional Hilbert spaces Skalarprodukträume and always have an orthonormal basis. For finite-dimensional vector spaces and separable Hilbert spaces one can find such using the Gram- Schmidt orthonormalization procedure. An example of an orthonormal basis, the standard basis (or canonical basis) of the three-dimensional space.
The term vector space can be generalized to that certain function spaces can be treated as vector spaces, and functions are then viewed as vectors. Two functions and a Skalarproduktraums are then called orthogonal if
Applies. For example, the L2 - scalar product for continuous real-valued functions on an interval is by
Defined. With respect to this scalar product, the two functions and to each other, for example, orthogonal on the interval, because it is
In full Skalarprodukträumen, so-called Hilbert spaces, so orthogonal polynomials and Orthogonalbasen be determined. However, many interesting spaces, such as L2 - spaces, infinite dimensional, see Hilbert space basis. In quantum mechanics the states of a system form a vector space, and accordingly we speak there also of orthogonal states.
A square orthogonal matrix is a real matrix, if it is compatible with the scalar product, in other words, if
Applies to all vectors. A matrix is orthogonal if and only if its columns (or their lines), regarded as vectors orthonormal to each other (not just orthogonal) are. Equivalent to the condition or. Orthogonal matrices describe rotations and reflections in the plane or in space. The set of all orthogonal matrices of size is the orthogonal group. The correspondence with matrices with complex entries is called unitary matrix.
Is a finite dimensional Euclidean vector space, then a linear map is called orthogonal if
Applies to all vectors. An orthogonal map thus receives angle between vectors and forms orthogonal vectors to orthogonal vectors from. A linear mapping is then exactly orthogonal if their matrix representation is an orthogonal matrix with respect to an orthonormal basis. Next is an orthogonal map is an isometry and thus also receives lengths and distances of vectors.
Orthogonal images are not to be confused with mutually orthogonal pictures. Here are pictures that are themselves regarded as vectors and their scalar product is zero.
Is a finite dimensional real or complex vector space with a scalar product, so there is any subspace projection along the orthogonal complement of which is called the orthogonal projection on. It is the uniquely determined linear mapping with the property that for all
- For all
Applies. If an infinite-dimensional Hilbert space, so this statement is in accordance with the projection set for closed subspaces. In this case, can be continuously selected.
Orthogonality is used in many applications, since this calculation can be carried out easier and more robust. Examples are:
- The Fourier transform and wavelet transform is performed in the signal processing
- QR factorization of matrices for solving eigenvalue problems
- The Gaussian quadrature for the numerical calculation of integrals
- Orthogonal arrays in the statistical design of experiments
- Orthogonal codes, such as the Walsh code in the channel coding
- The Orthogonalverfahren for measurement in geodesy