In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity to the linear algebra of bilinear forms.
Two elements and of a vector space with bilinear form
B
B(u,v)=0
The concept has been used in the context of orthogonal functions, orthogonal polynomials, and combinatorics.
V
\langleu,v\rangle
u\perpv
A
B
V
A
B
V
M
M*
m'
M*
m
M
\langlem',m\rangle=0
S'\subseteqM*
S\subseteqM
S'
S
A set of vectors in an inner product space is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set.
In certain cases, the word normal is used to mean orthogonal, particularly in the geometric sense as in the normal to a surface. For example, the y-axis is normal to the curve
y=x2
A vector space with a bilinear form generalizes the case of an inner product. When the bilinear form applied to two vectors results in zero, then they are orthogonal. The case of a pseudo-Euclidean plane uses the term hyperbolic orthogonality. In the diagram, axes x′ and t′ are hyperbolic-orthogonal for any given
\phi
In Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of 90° ( radians), or one of the vectors is zero.[2] Hence orthogonality of vectors is an extension of the concept of perpendicular vectors to spaces of any dimension.
The orthogonal complement of a subspace is the space of all vectors that are orthogonal to every vector in the subspace. In a three-dimensional Euclidean vector space, the orthogonal complement of a line through the origin is the plane through the origin perpendicular to it, and vice versa.[3]
Note that the geometric concept of two planes being perpendicular does not correspond to the orthogonal complement, since in three dimensions a pair of vectors, one from each of a pair of perpendicular planes, might meet at any angle.
In four-dimensional Euclidean space, the orthogonal complement of a line is a hyperplane and vice versa, and that of a plane is a plane.[3]
f
g
w
[a,b]
\langlef,g\ranglew=
| b | |
| \int | |
| a |
f(x)g(x)w(x)dx.
In simple cases,
w(x)=1
We say that functions
f
g
\langlef,g\ranglew=0.
Orthogonality of two functions with respect to one inner product does not imply orthogonality with respect to another inner product.
We write the norm with respect to this inner product as
\|f\|w=\sqrt{\langlef,f\ranglew}
The members of a set of functions
{fi\midi\inN
w
[a,b]
\langlefi,fj\ranglew=0\midi\nej.
w
[a,b]
\langlefi,fj\ranglew=\deltaij,
\deltaij=\left\{\begin{matrix}1,&&i=j\ 0,&&i ≠ j\end{matrix}\right.
In other words, every pair of them (excluding pairing of a function with itself) is orthogonal, and the norm of each is 1. See in particular the orthogonal polynomials.
(1,3,2)T,(3,-1,0)T,(1,3,-5)T
(1)(3)+(3)(-1)+(2)(0)=0 ,
(3)(1)+(-1)(3)+(0)(-5)=0 ,
(1)(1)+(3)(3)+(2)(-5)=0
(1,0,1,0,\ldots)T
(0,1,0,1,\ldots)T
| n | |
| Z | |
| 2 |
a
1\lek\lea-1
\begin{bmatrix}1&0&0&1&0&0&1&0\end{bmatrix}
\begin{bmatrix}0&1&0&0&1&0&0&1\end{bmatrix}
\begin{bmatrix}0&0&1&0&0&1&0&0\end{bmatrix}
2t+3
45t2+9t-17
1,\sin{(nx)},\cos{(nx)}\midn\inN
[0,2\pi],[-\pi,\pi]
2\pi
Various polynomial sequences named for mathematicians of the past are sequences of orthogonal polynomials. In particular:
[-1,1]
In combinatorics, two
n x n
n2
Two flat planes
A
B
A
B
A
B
O
A
B
O
A
B
In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a
(w,x,y,z)
(xy,xz,yz)
(wx,wy,wz)
xy
wz
xz
wy
yz
wx
More generally, two flat subspaces
S1
S2
M
N
S
M+N
S1
S2
\dim(S)=M+N
S1
S2
O
\dim(S)>M+N
S1
S2
\dim(S)=M+N
S1
S2
O