Everything about Orthogonal Matrix totally explained
In
matrix theory, a
real orthogonal matrix is a
square matrix Q whose
transpose is its
inverse:
»
An orthogonal matrix is a
special orthogonal matrix if its
determinant is +1:
»
Overview
An orthogonal matrix is the real specialization of a
unitary matrix, and thus always a
normal matrix. Although we consider only real matrices here, the definition can be used for matrices with entries from any
field. However, orthogonal matrices arise naturally from inner products, and for matrices of complex numbers that leads instead to the unitary requirement.
To see the inner product connection, consider a vector
v in an
n-dimensional real
inner product space. Written with respect to an orthonormal basis, the squared length of
v is
vTv. If a linear transformation, in matrix form
Qv, preserves vector lengths, then
»
Randomization
Some numerical applications, such as
Monte Carlo methods and exploration of high-dimensional data spaces, require generation of
uniformly distributed random orthogonal matrices. In this context, "uniform" is defined in terms of
Haar measure, which essentially requires that the distribution not change if multiplied by any freely chosen orthogonal matrix. It doesn't work to fill a matrix with
independent uniformly distributed random entries and then orthogonalize it. It
does work to fill it with independent
normally distributed random entries, then use QR decomposition. Stewart (1980) replaced this with a more efficient idea that Diaconis & Shahshahani (1987) later generalized as the "subgroup algorithm" (in which form it works just as well for permutations and rotations). To generate an (
n+1)×(
n+1) orthogonal matrix, take an
n×
n one and a uniformly distributed unit vector of dimension
n+1. Construct a Householder reflection from the vector, then apply it to the smaller matrix (embedded in the larger size with a 1 in the bottom corner).
Spin and Pin
A subtle technical problem afflicts some uses of orthogonal matrices. Not only are the group components with determinant +1 and −1 not connected to each other, even the +1 component,
SO(
n), isn't
simply connected (except for
SO(1), which is trivial). Thus it's sometimes advantageous, or even necessary, to work with a
covering group of
SO(
n), the
spin group, Spin(
n). Likewise,
O(
n) has covering groups, the
pin groups, Pin(
n). For
n > 2, Spin(
n) is simply connected, and thus the universal covering group for
SO(
n). By far the most famous example of a spin group is Spin(3), often seen in the form of unit
quaternions or
Pauli spin matrices.
In peculiarly
Ouroboros fashion, the Pin and Spin groups are found within
Clifford algebras, which themselves can be built from orthogonal matrices.
Rectangular matrices
If
Q is a rectangular matrix, then the conditions
QTQ =
I and
QQT =
I are not equivalent. The condition
QTQ =
I says that the columns of
Q are orthonormal. This can only happen if
Q is an
m×
n matrix with
n ≤
m. Similarly,
QQT =
I says that the rows of
Q are orthonormal, which requires
n ≥
m.
There is no standard terminology for these matrices. They are sometimes called "orthonormal matrices", sometimes "orthogonal matrices", and sometimes simply "matrices with orthonormal rows/columns".
Further Information
Get more info on 'Orthogonal Matrix'.
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