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In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
, two vectors in an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
are orthonormal if they are orthogonal (or perpendicular along a line)
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
s. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
is called an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
.


Intuitive overview

The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the
Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, two vectors are said to be ''perpendicular'' if the angle between them is 90° (i.e. if they form a
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
). This definition can be formalized in Cartesian space by defining the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
and specifying that two vectors in the plane are orthogonal if their dot product is zero. Similarly, the construction of the
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
of a vector is motivated by a desire to extend the intuitive notion of the
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
of a vector to higher-dimensional spaces. In Cartesian space, the ''norm'' of a vector is the square root of the vector dotted with itself. That is, :\, \mathbf \, = \sqrt Many important results in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
deal with collections of two or more orthogonal vectors. But often, it is easier to deal with vectors of
unit length Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (al ...
. That is, it often simplifies things to only consider vectors whose norm equals 1. The notion of restricting orthogonal pairs of vectors to only those of unit length is important enough to be given a special name. Two vectors which are orthogonal and of length 1 are said to be ''orthonormal''.


Simple example

What does a pair of orthonormal vectors in 2-D Euclidean space look like? Let u = (x1, y1) and v = (x2, y2). Consider the restrictions on x1, x2, y1, y2 required to make u and v form an orthonormal pair. * From the orthogonality restriction, u • v = 0. * From the unit length restriction on u, , , u, , = 1. * From the unit length restriction on v, , , v, , = 1. Expanding these terms gives 3 equations: #x_1 x_2 + y_1 y_2 = 0 \quad #\sqrt = 1 #\sqrt = 1 Converting from Cartesian to
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
, and considering Equation (2) and Equation (3) immediately gives the result r1 = r2 = 1. In other words, requiring the vectors be of unit length restricts the vectors to lie on the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
. After substitution, Equation (1) becomes \cos \theta _1 \cos \theta _2 + \sin \theta _1 \sin \theta _2 = 0. Rearranging gives \tan \theta _1 = - \cot \theta _2. Using a
trigonometric identity In trigonometry, trigonometric identities are Equality (mathematics), equalities that involve trigonometric functions and are true for every value of the occurring Variable (mathematics), variables for which both sides of the equality are defined. ...
to convert the cotangent term gives : \tan ( \theta_1 ) = \tan \left( \theta_2 + \tfrac \right) : \Rightarrow \theta _1 = \theta _2 + \tfrac It is clear that in the plane, orthonormal vectors are simply radii of the unit circle whose difference in angles equals 90°.


Definition

Let \mathcal be an
inner-product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. A set of vectors : \left\ \in \mathcal is called orthonormal
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
: \forall i,j : \langle u_i , u_j \rangle = \delta_ where \delta_ \, is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
and \langle \cdot , \cdot \rangle is the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
defined over \mathcal.


Significance

Orthonormal sets are not especially significant on their own. However, they display certain features that make them fundamental in exploring the notion of
diagonalizability In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
of certain
operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
on vector spaces.


Properties

Orthonormal sets have certain very appealing properties, which make them particularly easy to work with. *Theorem. If is an orthonormal list of vectors, then \forall \textbf := _1, \cdots, a_n \ \, a_1 \textbf_1 + a_2 \textbf_2 + \cdots + a_n \textbf_n\, ^2 = , a_1, ^2 + , a_2, ^2 + \cdots + , a_n, ^2 *Theorem. Every orthonormal list of vectors is
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
.


Existence

* Gram-Schmidt theorem. If is a linearly independent list of vectors in an inner-product space \mathcal, then there exists an orthonormal list of vectors in \mathcal such that ''span''(e1, e2,...,en) = ''span''(v1, v2,...,vn). Proof of the Gram-Schmidt theorem is
constructive Although the general English usage of the adjective constructive is "helping to develop or improve something; helpful to someone, instead of upsetting and negative," as in the phrase "constructive criticism," in legal writing ''constructive'' has ...
, and discussed at length elsewhere. The Gram-Schmidt theorem, together with the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, guarantees that every vector space admits an orthonormal basis. This is possibly the most significant use of orthonormality, as this fact permits
operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
on inner-product spaces to be discussed in terms of their action on the space's orthonormal basis vectors. What results is a deep relationship between the diagonalizability of an operator and how it acts on the orthonormal basis vectors. This relationship is characterized by the
Spectral Theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful ...
.


Examples


Standard basis

The
standard basis In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the c ...
for the
coordinate space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
F''n'' is : Any two vectors ei, ej where i≠j are orthogonal, and all vectors are clearly of unit length. So forms an orthonormal basis.


Real-valued functions

When referring to
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
-valued functions, usually the inner product is assumed unless otherwise stated. Two functions \phi(x) and \psi(x) are orthonormal over the interval ,b/math> if :(1)\quad\langle\phi(x),\psi(x)\rangle = \int_a^b\phi(x)\psi(x)dx = 0,\quad :(2)\quad, , \phi(x), , _2 = , , \psi(x), , _2 = \left \phi(x), ^2dx\right\frac = \left \psi(x), ^2dx\right\frac = 1.


Fourier series

The
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
is a method of expressing a periodic function in terms of sinusoidal
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
functions. Taking C π,πto be the space of all real-valued functions continuous on the interval π,πand taking the inner product to be :\langle f, g \rangle = \int_^ f(x)g(x)dx it can be shown that :\left\, \quad n \in \mathbb forms an orthonormal set. However, this is of little consequence, because C π,πis infinite-dimensional, and a finite set of vectors cannot span it. But, removing the restriction that ''n'' be finite makes the set
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in C π,πand therefore an orthonormal basis of C π,π


See also

*
Orthogonalization In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors in an inner product space (most commonly the Euclidean spa ...


Sources

* * {{Citation , last1=Chen , first1=Wai-Kai , title=Fundamentals of Circuits and Filters , publisher=
CRC Press The CRC Press, LLC is an American publishing group that specializes in producing technical books. Many of their books relate to engineering, science and mathematics. Their scope also includes books on business, forensics and information tec ...
, location= Boca Raton , edition=3rd , pag
62
isbn=978-1-4200-5887-1 , year=2009 Linear algebra Functional analysis