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 matrice ...
, two
vectors in an
inner product space are orthonormal if they are
orthogonal (or perpendicular along a line)
unit vectors. 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.
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,
:
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 matrice ...
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'' (a ...
. 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 = (x
1, y
1) and v = (x
2, y
2).
Consider the restrictions on x
1, x
2, y
1, y
2 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:
#
#
#
Converting from Cartesian to
polar coordinates, and considering Equation
and Equation
immediately gives the result r
1 = r
2 = 1. In other words, requiring the vectors be of unit length restricts the vectors to lie on the
unit circle.
After substitution, Equation
becomes
. Rearranging gives
. Using a
trigonometric identity to convert the
cotangent term gives
:
:
It is clear that in the plane, orthonormal vectors are simply radii of the unit circle whose difference in angles equals 90°.
Definition
Let
be an
inner-product space. A set of vectors
:
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 bic ...
:
where
is the
Kronecker delta and
is the
inner product defined over
.
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 is an orthonormal list of vectors, then
*Theorem. Every orthonormal list of vectors is
linearly independent.
Existence
*
Gram-Schmidt theorem. If is a linearly independent list of vectors in an inner-product space
, then there exists an orthonormal list of vectors in
such that ''span''(e
1, e
2,...,e
n) = ''span''(v
1, v
2,...,v
n).
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, guarantees that every vector space admits an orthonormal basis. This is possibly the most significant use of orthonormality, as this fact permits
operators 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.
Examples
Standard basis
The
standard basis for the
coordinate space F
''n'' is
:
Any two vectors e
i, e
j 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-valued
functions, usually the
L² inner product is assumed unless otherwise stated. Two functions
and
are orthonormal over the
interval