In linear algebra, an orthogonal matrix is a square matrix G whose transpose is its inverse, i.e., 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. ... See transposition for meanings of this term in telecommunication and music. ... 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. ...

GG^T = G^TG = I_n.

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. 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 associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... The text or formatting below is generated by a template which has been proposed for deletion. ...

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. In mathematics, an orthonormal basis of an inner product space V(i. ... In mathematics, the dot product (also known as the scalar product and the inner product) is a sesquilinear function (·) : V × V → F, where V is a vector space over the field F, having some further properties. ...

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 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. ... This article is about angles in geometry. ... In general English usage, length (symbol: 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) is...

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 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. ...

1 = det(I) = det(GG^T) = det(G)det(G^T) = (det(G))².

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). This article is about rotation as a movement of a physical body. ... 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. ... The graph of the absolute value function In mathematics, the absolute value (or modulus) of a real number is its numerical value without regard to its sign. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

If Q is orthogonal, then one can always find an orthogonal matrix P such that

where the matrices R₁,...,R_k are 2-by-2 rotation matrices. Intuitively, this result means that every orthogonal matrix describes a combination of rotations and reflections. The matrices R₁,...,R_k correspond to the non-real eigenvalues of Q.

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 a 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...

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. To 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.

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:

1. they are orthogonal matrices as defined in this article
2. 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. ...

See also

• orthogonal group
• unitary matrix
• symplectic matrix
• skew-symmetric matrix