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Ornstein-Uhlenbeck process A Gaussian, Thence the inverse of an orthogonal matrix coin-
Markov stochastic processes defined by a linear cides with its transpose. Example: Rotations in
3
stochastic differential equation R (as well as reflections) are elements of O(3).
Analogously, the orthogonal group O(r, s)
dX =−bXdt + adW(t) (with r + s = m) is the group of isometries of
m
the indefinite metric of signature (r, s) on R .
in which a, b > 0, and W(t) is the Wiener pro- Example: The Lorentz group is the orthogonal
cess, i.e., dW/dt is the white noise. group O(1,m − 1).
The subgroup of isometries α : R m → R m
orthogonal Describing two elements V , V 2 which preserve the orientation of (R ,η) is
m
1
of a Hilbert space such that /v ,v 0= 0. denoted by SO(r, s), and it is called the special
2
1
orthogonal group.
orthogonal group The group O(m) is the
m
group of isometries of R with the standard posi- orthonormal basis An orthonormal set
tive definite metric δ. Let us choose an ortho- which is also a basis. Synonym: complete
m
normal basis E in R , the metric δ is expressed orthonormal set.
i
i
j
i
by δ = δ E ⊗ E (where E is the dual
ij
m
m
basis). An endomorphism α : R → R is an orthonormal set A set {u } of a Hilbert
α
element of the orthogonal group O(m) if and space H satisfying
only if

0 if α = β
j
b
a
A δ ab A = δ ij α(E ) = A E . /u ,u 0= .
β
i
α
j
i
j
i
1 if α = β

© 2003 by CRC Press LLC
© 2003 by CRC Press LLC