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# Orthogonal matrix

In linear algebra, an orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e. where I is the identity matrix. [1]

105 relations: Absolute value, Alternating group, Angular velocity, Axis–angle representation, Charles F. Van Loan, Cholesky decomposition, Clifford algebra, Compact space, Complex number, Condition number, Connected space, Coset, Covering space, Determinant, Diagonal matrix, Diagonalizable matrix, Dimension (vector space), Direct product of groups, Discrete cosine transform, Dot product, Eigendecomposition of a matrix, Eigenvalues and eigenvectors, Euclidean space, Euler angles, Exact sequence, Factorial, Fiber bundle, Field (mathematics), Gaussian elimination, Gene H. Golub, Generalized inverse, Givens rotation, Gram–Schmidt process, Group (mathematics), Haar measure, Householder transformation, Identity matrix, If and only if, Improper rotation, Independence (probability theory), Index of a subgroup, Invertible matrix, Involutory matrix, Isometry, Jacobi rotation, Lamar University, Lie algebra, Lie group, Linear algebra, Linear least squares (mathematics), ... Expand index (55 more) »

## Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

## Alternating group

In mathematics, an alternating group is the group of even permutations of a finite set.

## Angular velocity

In physics, the angular velocity of a particle is the rate at which it rotates around a chosen center point: that is, the time rate of change of its angular displacement relative to the origin.

## Axis–angle representation

In mathematics, the axis–angle representation of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector indicating the direction of an axis of rotation, and an angle describing the magnitude of the rotation about the axis.

## Charles F. Van Loan

Charles Francis Van Loan is an emeritus professor of computer science and the Joseph C. Ford Professor of Engineering at Cornell University,.

## Cholesky decomposition

In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced /ʃ-/) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g. Monte Carlo simulations.

## Clifford algebra

In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra.

## Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Condition number

In the field of numerical analysis, the condition number of a function with respect to an argument measures how much the output value of the function can change for a small change in the input argument.

## Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.

## Coset

In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then Only when H is normal will the set of right cosets and the set of left cosets of H coincide, which is one definition of normality of a subgroup.

## Covering space

In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.

## Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

## Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.

## Diagonalizable matrix

In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P−1AP is a diagonal matrix.

## Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.

## Direct product of groups

In group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted.

## Discrete cosine transform

A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies.

## Dot product

In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.

## Eigendecomposition of a matrix

In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors.

## Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Euler angles

The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.

## Exact sequence

An exact sequence is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry.

## Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The value of 0! is 1, according to the convention for an empty product.

## Fiber bundle

In mathematics, and particularly topology, a fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally may have a different topological structure.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## Gaussian elimination

In linear algebra, Gaussian elimination (also known as row reduction) is an algorithm for solving systems of linear equations.

## Gene H. Golub

Gene Howard Golub (February 29, 1932 – November 16, 2007), Fletcher Jones Professor of Computer Science (and, by courtesy, of Electrical Engineering) at Stanford University, was one of the preeminent numerical analysts of his generation.

## Generalized inverse

In mathematics, and in particular, algebra, a generalized inverse of an element x is an element y that has some properties of an inverse element but not necessarily all of them.

## Givens rotation

In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes.

## Gram–Schmidt process

In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space Rn equipped with the standard inner product.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Haar measure

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.

## Householder transformation

In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin.

## Identity matrix

In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.

## If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

## Improper rotation

In geometry, an improper rotation,.

## Independence (probability theory)

In probability theory, two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of the other.

## Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G: H| or or (G:H).

## Invertible matrix

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.

## Involutory matrix

In mathematics, an involutory matrix is a matrix that is its own inverse.

## Isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

## Jacobi rotation

In numerical linear algebra, a Jacobi rotation is a rotation, Qkℓ, of a 2-dimensional linear subspace of an n-dimensional inner product space, chosen to zero a symmetric pair of off-diagonal entries of an n×n real symmetric matrix, A, when applied as a similarity transformation: \begin \end \to \begin \end.

## Lamar University

Lamar University, often referred to as Lamar or LU, is a public coeducational doctoral/research university in Beaumont, Texas.

## Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.

## Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

## Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

## Linear least squares (mathematics)

In statistics and mathematics, linear least squares is an approach to fitting a mathematical or statistical model to data in cases where the idealized value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model.

## Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

## Massachusetts Institute of Technology

The Massachusetts Institute of Technology (MIT) is a private research university located in Cambridge, Massachusetts, United States.

## Matrix decomposition

In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices.

## Matrix exponential

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function.

## Matrix multiplication

In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.

## Matrix norm

In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).

## Monte Carlo method

Monte Carlo methods (or Monte Carlo experiments) are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results.

## MP3

MP3 (formally MPEG-1 Audio Layer III or MPEG-2 Audio Layer III) is an audio coding format for digital audio.

## Newton's method

In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.

## Nicholas Higham

Nicholas John Higham FRS (born 25 December 1961 in Salford) is a British numerical analyst and Richardson Professor of Applied Mathematics at the School of Mathematics at the University of Manchester.

## Normal distribution

In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution.

## Normal matrix

In mathematics, a complex square matrix is normal if where is the conjugate transpose of.

## Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.

## Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).

## Numerical stability

In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms.

## Orthogonal group

In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.

## Orthogonal Procrustes problem

The orthogonal Procrustes problem is a matrix approximation problem in linear algebra.

## Orthogonality

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.

## Orthogonalization

In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace.

## Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.

## Orthonormality

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and unit vectors.

## Overdetermined system

In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns.

## Parity of a permutation

In mathematics, when X is a finite set of at least two elements, the permutations of X (i.e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations.

\pi.

## Persi Diaconis

Persi Warren Diaconis (born January 31, 1945) is an American mathematician of Greek descent and former professional magician.

## Pin group

In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space.

## Plane of rotation

In geometry, a plane of rotation is an abstract object used to describe or visualize rotations in space.

## Point group

In geometry, a point group is a group of geometric symmetries (isometries) that keep at least one point fixed.

## Point reflection

In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) is a type of isometry of Euclidean space.

## Polar decomposition

In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to the polar form of a nonzero complex number z as z.

## QR decomposition

In linear algebra, a QR decomposition (also called a QR factorization) of a matrix is a decomposition of a matrix A into a product A.

## Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers.

## Quotient group

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.

## Reflection (mathematics)

In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.

## Reflection group

In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space.

## Rotation

A rotation is a circular movement of an object around a center (or point) of rotation.

## Rotation (mathematics)

Rotation in mathematics is a concept originating in geometry.

## Rotation matrix

In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space.

## Semidirect product

In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product.

## Simply connected space

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.

## Singular-value decomposition

In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix.

## Skew-symmetric matrix

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative; that is, it satisfies the condition In terms of the entries of the matrix, if aij denotes the entry in the and; i.e.,, then the skew-symmetric condition is For example, the following matrix is skew-symmetric: 0 & 2 & -1 \\ -2 & 0 & -4 \\ 1 & 4 & 0\end.

## Spectral theorem

In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis).

## Spin group

In mathematics the spin group Spin(n) is the double cover of the special orthogonal group, such that there exists a short exact sequence of Lie groups (with) As a Lie group, Spin(n) therefore shares its dimension,, and its Lie algebra with the special orthogonal group.

## Square matrix

In mathematics, a square matrix is a matrix with the same number of rows and columns.

## Square root of a matrix

In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices.

## Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

## Symmetric group

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.

## Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.

## Symplectic matrix

In mathematics, a symplectic matrix is a 2n×2n matrix M with real entries that satisfies the condition where MT denotes the transpose of M and Ω is a fixed 2n×2n nonsingular, skew-symmetric matrix.

## Transpose

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT (also written A′, Atr, tA or At).

## Uniform distribution (continuous)

In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable.

## Unit vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.

## Unitary matrix

In mathematics, a complex square matrix is unitary if its conjugate transpose is also its inverse—that is, if where is the identity matrix.

## Unitary transformation

In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.

## References

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