Orthogonal matrix Guide, Meaning , Facts, Information and Description
In
linear algebra, an
orthogonal matrix is a
square matrix G whose
transpose is its
inverse, i.e.,
This definition can be given for matrices with entries from any
field, but the most common case is the one of matrices with
real entries, and only that case will be considered in the rest of this article.
A real square matrix is orthogonal if and only if its columns form an orthonormal basis of Rn with the ordinary Euclidean dot product, which is the case if and only if its rows form an orthonormal basis of Rn.
Geometrically, orthogonal matrices describe linear transformations of Rn which preserve angles and lengths, such as rotations and reflections. They are compatible with the Euclidean inner product in the following sense: if G is orthogonal and x and y are vectors in Rn, then
Conversely, if
V is any finite-dimensional real
inner product space and
f :
V →
V is a linear map with
for all elements
x,
y of
V, then
f is described by an orthogonal matrix with respect to any orthonormal basis of
V.
The inverse of every orthogonal matrix is again orthogonal, as is the matrix product of two orthogonal matrices. This shows that the set of all n×n orthogonal matrices forms a group. It is a Lie group of dimension n(n − 1)/2 and is called the orthogonal group, denoted by O(n).
The determinant of any orthogonal matrix is 1 or −1. That can be shown as follows:
In three dimensions, the orthogonal matrices with determinant 1 correspond to proper rotations and those with determinant −1 to
improper rotations.
The set of all orthogonal matrices whose determinant is 1 is a
subgroup of O(
n) of
index 2, the
special orthogonal group SO(
n).
All eigenvalues of an orthogonal matrix, even the complex ones, have absolute value 1. Eigenvectors for different eigenvalues are orthogonal.
If Q is orthogonal, then one can always find an orthogonal matrix P such that
where the matrices
R1,...,
Rk are 2-by-2 rotation matrices. Intuitively, this result means that every orthogonal matrix describes a combination of rotations and reflections.
The matrices
R1,...,
Rk correspond to the non-real eigenvalues of
Q.
If A is an arbitrary m-by-n matrix of rank n, we can always write
where
Q is an orthogonal
m-by-
m matrix and
R is an upper triangular
n-by-
n matrix with positive main diagonal entries. This is known as a
QR decomposition of
A and can be proven by
applying the
Gram-Schmidt process to the columns of
A. It is useful for numerically solving
systems of linear equations and
least squares problems.
The complex analog to orthogonal matrices are the unitary matrices.
Matrix representation of Clifford algebras
This is meant as a simple introduction.
There is a second geometrical meaning for orthogonal matrices.
In matrix representations of Clifford algebras some of them are regarded as base vectors. Let me give a simple example.
Normally in R2 we have the basic vectors e1 = [1 0] and e2 =[0 1], so that a point in this plane is
- [x y]= x·[1 0] + y·[0 1]
The orthogonal matrix
-
represents a reflection around the bisecting line because the two basic vectors get exchanged.
The orthogonal matrix
represents a
reflection in the
x-axis because the point [x y] has [x,−y] as image.
These two reflections
anticommute (the result changes sign if the order is reversed)
-
This is a rotation
If we now no longer regard them as linear transformations but as basic
vectors for a 2D plane.
A point with coordinates (x,y) would in this plane be represented by the matrix
The square of this matrix is the square of its
norm (the inner product with itself)
If we now define the inner product as
-
because the base vectors anticommute we see that
The matrices e
1 and e
2
- are orthogonal in both senses:
they are orthogonal matrices as defined in this article
they represent orthogonal basicvectors (a right angle between them) because they anticommute.
See more at
representations of Clifford algebras.
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