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In linear algebra, an orthogonal matrix is a square matrix G whose transpose is its inverse, Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...
For the square matrix section, see square matrix. ...
In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...
In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...
The 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. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
Examples
Below are several examples of orthogonal matrices and their corresponding linear transformation. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
- (the identity transformation)
- (counterclockwise rotation by angle θ)
- (reflection across the x-axis)
- (rotoinversion with axis (0,−3/5,4/5), angle π/2)
Geometric interpretation In 1D the orthogonal matrices represent the identity and the reflection in the origin.
In 2D the orthogonal matrices represent the identity, the rotations about the origin, and the reflections in a line through the origin.
In 3D the orthogonal matrices represent:
- the identity
- the rotations about a line through the origin
- the reflections in a plane through the origin
- the rotations about a line through the origin combined with a reflection in the plane through the origin that is perpendicular to that line.
Each rotation can be decomposed into three about fixed axes, in terms of flight dynamics pitch, roll and yaw. Flight dynamics is the study of orientation of air and space vehicles and how to control the critical flight parameters, typically named pitch, roll and yaw. ...
In 4D one example is a change of x and y like a rotation, keeping z and the fourth coordinate u fixed. For example, the point (5,0,3,8) is mapped to (4,3,3,8). The equivalent in 4D of an axis of rotation is a plane, in this case the zu-plane. The distance of the point to the "axis" is that to the nearest point on it, in this case (0,0,3,8), so the distance is 5. The point moves in a plane orthogonal to the "axis" (the plane given by z=3, u=8), keeping the distance to (0,0,3,8) the same. The axis of rotation of a rotating body is a line such that the distance between any point on the line and any point of the body remains constant under the rotation. ...
As a result of what is explained in the next section, a 4D isometry can also be a combination of two such "rotations", with "axes" which are orthogonal planes (as opposed to what applies in 3D, these two rotations are not equivalent to one combined rotation). Also one such "rotation" can be combined with a reflection. See also SO(4). SO(4) is the symbol used in mathematics for the group of rotations about a fixed point in four-dimensional Euclidean space (for short, the 4D rotation group). ...
The identity can be considered a rotation by a zero angle.
Properties A real square matrix is orthogonal if and only if its columns form an orthonormal basis of the Euclidean space Rn with the ordinary Euclidean dot product, which is the case if and only if its rows form an orthonormal basis of Rn. The definition is slightly inconsistent: the matrix is called orthogonal if the columns are orthonormal (if the columns of G are orthogonal but not orthonormal, then GTG is diagonal). In mathematics, an orthonormal basis of an inner product space V(i. ...
In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...
In linear algebra, a diagonal matrix is a square matrix in which only the entries in the main diagonal are non-zero. ...
Geometrically, orthogonal matrices describe isometries which keep the origin fixed, that is, linear transformations of Rn which preserve lengths, and hence also angles. Such a transformation is called an orthogonal transformation. Orthogonal transformations include rotations, reflections and any combination of them. 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 In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ...
In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
In general English usage, length (symbols: l, L) is but one particular instance of distance – an objects length is how long the object is – but in the physical sciences and engineering, the word length is in some contexts used synonymously with distance. Height is vertical distance; width (or breadth...
This article is about angles in geometry. ...
Rotation is the movement of a body in such a way that the distance between a certain fixed point and any given point of that body remains constant. ...
IT IS KNOWN AS MARK a lunitice insain int gw brain ...
Conversely, if V is any finite-dimensional real inner product space and f : V → V is a linear map with In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to introduce geometrical notions such as angles and lengths of vectors. ...
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). In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
In mathematics, a Lie group (IPA ) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...
In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...
The determinant of any orthogonal matrix is 1 or −1. That can be shown as follows: In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
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). Rotation is the movement of a body in such a way that the distance between a certain fixed point and any given point of that body remains constant. ...
In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). ...
In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation...
In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...
In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...
Orthogonal matrices preserve lengths and hence, all eigenvalues have absolute value 1, i.e., they are on the unit circle centered at 0 in the complex plane. Eigenvectors for different eigenvalues are orthogonal. In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...
In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ...
In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...
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. We see here confirmed that if n is odd, there is at least one real eigenvalue, 1 or -1.
If A is an arbitrary m-by-n matrix of rank n, we can always write In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ...
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. In linear algebra, the QR decomposition of a matrix is a factorization expressing as where is an orthogonal matrix (), and is an upper triangular matrix. ...
In mathematics and numerical analysis, the Gram-Schmidt process of linear algebra is a method of orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn. ...
In mathematics and linear algebra, a system of linear equations is a set of linear equations such as 3x1 + 2x2 − x3 = 1 2x1 − 2x2 + 4x3 = −2 −x1 + ½x2 − x3 = 0. ...
Least squares is a mathematical optimization technique that attempts to find a best fit to a set of data by attempting to minimize the sum of the squares of the differences (called residuals) between the fitted function and the data. ...
The complex analog to orthogonal matrices are the unitary matrices. In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse...
Orthogonal matrices in numerical analysis Orthogonal matrices are important in numerical analysis because their condition number is one (which is the minimum). This means that errors are not magnified when multiplying with an orthogonal matrix: such a computation is numerical stable. Many algorithms use orthogonal matrices like Householder reflections and Givens rotations for this reason. Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ...
In numerical analysis, the condition number associated with a numerical problem is a measure of that quantitys amenability to digital computation, that is, how well-posed the problem is. ...
In the mathematical subfield of numerical analysis, numerical stability is a property of numerical algorithms. ...
In mathematics, a Householder transformation in 3-dimensional space is the reflection of a vector in a plane. ...
In mathematics, a Givens rotation is a matrix of the form where c = cos(θ) and s = sin(θ) appear in the i-th / k-th row and column, respectively. ...
Algorithm for adjusting a matrix so that it is orthogonal In numerical computations the product of two orthogonal matrices may not be orthogonal due to floating point error. The following algorithm will clean up a nearly orthogonal matrix (assuming that the columns in the matrix are unit vectors): A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ...
Flowcharts are often used to represent algorithms. ...
In mathematics, a unit vector in a normed vector space is a vector (most commonly a spatial vector) whose length is 1. ...
for col1 = first column to last column for col2 = col1+1 to last column let dp = dot product of col1 and col2 for row = 1 to num of rows mtx(row, col2) -= mtx(row, col1) * dp let len = sqrt(dot product of col2 and col2) for row = 1 to num of rows mtx(row, col2) /= len The first inner loop moves col2 so that it is at right angles to col1, by moving the tip of the vector opposite to the direction of col1 by the amount of component that col2 has in col1's direction. Doing this never disturbs prior adjustments, because col1 and col2 are always at right angles to all prior columns. This will shorten col2, so we have to re-normalise it.
This algorithm is essentially a coding of the Gram-Schmidt process for orthogonalization of the columns. It is not the most reliable, nor the most efficient, nor the most invariant. The polar decomposition factors a matrix into a pair, one of which is the unique closest orthogonal matrix to the given matrix in the Frobenius norm (or one of the closest if the given matrix is singular). For a near-orthogonal matrix, rapid convergence to true orthogonality is often achieved by a "Newton's method" approach due to Higham, repeatedly averaging the matrix with its inverse transpose. Dubrulle has published an accelerated method with a convenient convergence test. In mathematics and numerical analysis, the Gram-Schmidt process of linear algebra is a method of orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn. ...
In linear algebra, orthogonalization means the following: we start with vectors v1,...,vk in an inner product space, most commonly the Euclidean space Rn which are linearly independent and we want to find mutually orthogonal vectors u1,...,uk which generate the same subspace as the vectors v1,...,vk. ...
In mathematics, particularly in linear algebra and functional analysis, the polar decomposition is a canonical factorization of any linear mapping T between complex Hilbert spaces as the product of a partial isometry and a non-negative self-adjoint operator. ...
In mathematics, the term matrix norm can have two meanings: A vector norm on matrices, i. ...
In numerical analysis, Newtons method (or the Newton-Raphson method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ...
For example, the simple averaging algorithm takes seven steps
which acceleration trims to two steps (with γ = 0.353553, 0.565685).
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. In mathematics, the representations of Clifford algebras are also known as Clifford modules. ...
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 word reflection (also spelt reflexion in British English) can refer to several different concepts: In mathematics, reflection is the transformation of a space. ...
The orthogonal matrix represents a reflection in the x-axis because the point [x y] has [x,−y] as image. The word reflection (also spelt reflexion in British English) can refer to several different concepts: In mathematics, reflection is the transformation of a space. ...
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) The word norm coming from the latin word norma which means angle measure or (lawlike) rule, has a number of meanings: A social or sociological norm; see norm (sociology). ...
If we now define the inner product as because the base vectors anticommute we see that The matrices e1 and e2 - 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. In mathematics, the representations of Clifford algebras are also known as Clifford modules. ...
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