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orthonormality
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{{Short description|Property of two or more vectors that are orthogonal and of unit length}}In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpendicular to each other. 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 is called an orthonormal basis.- the content below is remote from Wikipedia
- it has been imported raw for GetWiki
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, two vectors are said to be perpendicular if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero.Similarly, the construction of the norm of a vector is motivated by a desire to extend the intuitive notion of the length 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{x} | = sqrt{ mathbf{x} cdot mathbf{x}}
Many important results in linear algebra deal with collections of two or more orthogonal vectors. But often, it is easier to deal with vectors of unit length. 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.
- x_1 x_2 + y_1 y_2 = 0 quad
- sqrt{{x_1}^2 + {y_1}^2} = 1
- sqrt{{x_2}^2 + {y_2}^2} = 1
tan ( theta_1 ) = tan left( theta_2 + tfrac{pi}{2} right)
Rightarrow theta _1 = theta _2 + tfrac{pi}{2}
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{V} be an inner-product space. A set of vectors
left{ u_1 , u_2 , ldots , u_n , ldots right} in mathcal{V}
is called orthonormal if and only if
forall i,j : langle u_i , u_j rangle = delta_{ij}
where delta_{ij} , is the Kronecker delta and langle cdot , cdot rangle is the inner product defined over mathcal{V}.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 of certain operators on vector spaces.Properties
Orthonormal sets have certain very appealing properties, which make them particularly easy to work with.- Theorem. If {e1, e2, ..., en} is an orthonormal list of vectors, then forall textbf{a} := [a_1, cdots, a_n]; |a_1 textbf{e}_1 + a_2 textbf{e}_2 + cdots + a_n textbf{e}_n|^2 = |a_1|^2 + |a_2|^2 + cdots + |a_n|^2
- Theorem. Every orthonormal list of vectors is linearly independent.
Existence
- Gram-Schmidt theorem. If {v1, v2,...,vn} is a linearly independent list of vectors in an inner-product space mathcal{V}, then there exists an orthonormal list {e1, e2,...,en} of vectors in mathcal{V} such that span(e1, e2,...,en) = span(v1, v2,...,vn).
Examples
Standard basis
The standard basis for the coordinate space Fn is
{|
Any two vectors ei, ej where iâ j are orthogonal, and all vectors are clearly of unit length. So {e1, e2,...,en} forms an orthonormal basis.
vdots}} |
Real-valued functions
When referring to real-valued functions, usually the L² inner product is assumed unless otherwise stated. Two functions phi(x) and psi(x) are orthonormal over the interval [a,b] if
(1)quadlanglephi(x),psi(x)rangle = int_a^bphi(x)psi(x)dx = 0,quad{rm and}
(2)quad||phi(x)||_2 = ||psi(x)||_2 = left[int_a^b|phi(x)|^2dxright]^frac{1}{2} = left[int_a^b|psi(x)|^2dxright]^frac{1}{2} = 1.
Fourier series
The Fourier series is a method of expressing a periodic function in terms of sinusoidal basis functions.Taking C[âÏ,Ï] to be the space of all real-valued functions continuous on the interval [âÏ,Ï] and taking the inner product to be
langle f, g rangle = int_{-pi}^{pi} f(x)g(x)dx
it can be shown that
left{ frac{1}{sqrt{2pi}}, frac{sin(x)}{sqrt{pi}}, frac{sin(2x)}{sqrt{pi}}, ldots, frac{sin(nx)}{sqrt{pi}}, frac{cos(x)}{sqrt{pi}}, frac{cos(2x)}{sqrt{pi}}, ldots, frac{cos(nx)}{sqrt{pi}} right}, quad n in mathbb{N}
forms an orthonormal set.However, this is of little consequence, because C[âÏ,Ï] is infinite-dimensional, and a finite set of vectors cannot span it. But, removing the restriction that n be finite makes the set dense in C[âÏ,Ï] and therefore an orthonormal basis of C[âÏ,Ï].See also
Sources
- {{Citation | last1=Axler | first1=Sheldon | author-link=Sheldon Axler| title=Linear Algebra Done Right | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | page= 106â110| isbn=978-0-387-98258-8 | year=1997}}
- {{Citation | last1=Chen | first1=Wai-Kai | title=Fundamentals of Circuits and Filters | publisher=CRC Press | location=Boca Raton | edition=3rd | page=62| isbn=978-1-4200-5887-1 | year=2009}}
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