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Orthogonal transformation

In linear algebra, an orthogonal transformation is a linear transformation T: V \to V on a real inner product space V that preserves the inner product, meaning \langle Tv, Tw \rangle = \langle v, w \rangle for all vectors v, w \in V.^[1] Equivalently, it preserves the Euclidean norm of vectors (\|Tv\| = \|v\|) and the angles between them, ensuring that distances and orientations are maintained under the mapping.^[2]^[3] When represented in a standard orthonormal basis, an orthogonal transformation corresponds to an orthogonal matrix Q, a square matrix satisfying Q^T Q = I_n, where Q^T is the transpose and I_n is the n \times n identity matrix; this implies Q^{-1} = Q^T.^[2]^[4] The columns (and rows) of Q form an orthonormal basis for \mathbb{R}^n, and the determinant of Q is either +1 (for proper rotations) or -1 (for improper rotations involving a reflection).^[1]^[2] Orthogonal transformations form the orthogonal group O(n), a Lie group under matrix multiplication, consisting of all such matrices; the subgroup of proper rotations is the special orthogonal group SO(n).^[1] Common examples include rotations in the plane, given by matrices like \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}, and reflections across a hyperplane, both of which preserve vector lengths and orthogonality.^[2]^[3]^[4] These transformations are fundamental in various fields, including physics for describing rigid body motions and coordinate changes, computer graphics for rotations and reflections, and numerical methods like QR decomposition where the Q factor is orthogonal to stabilize computations.^[2] They also underpin signal processing techniques, such as the discrete Fourier transform, by enabling efficient, norm-preserving representations of data.^[2]

Definition and Properties

Definition

An orthogonal transformation is a linear transformation T: V \to V on a finite-dimensional real inner product space V that satisfies \langle T(\mathbf{u}), T(\mathbf{v}) \rangle = \langle \mathbf{u}, \mathbf{v} \rangle for all \mathbf{u}, \mathbf{v} \in V.^[5]^[6] This preservation of the inner product is equivalent to the transformation preserving the Euclidean norm, since \|T(\mathbf{v})\|^2 = \langle T(\mathbf{v}), T(\mathbf{v}) \rangle = \langle \mathbf{v}, \mathbf{v} \rangle = \|\mathbf{v}\|^2 for all \mathbf{v} \in V.^[6]^[5] While the primary focus is on real inner product spaces, the concept extends to complex spaces through unitary transformations, which preserve the Hermitian inner product in an analogous manner.^[6] The term originated in the context of 19th-century geometry, notably through the work of mathematicians such as Augustin-Louis Cauchy, who employed such transformations in analyzing symmetric matrices.^[7] These transformations relate to isometries of Euclidean space, as preserving the inner product implies preserving distances between points.^[6]

Key Properties

Orthogonal transformations on a real inner product space preserve the inner product, which implies they preserve the Euclidean norm of vectors, ensuring that lengths remain unchanged under the transformation.^[1] This norm preservation directly leads to the invertibility of orthogonal transformations, as the preservation of lengths guarantees that the transformation is bijective and thus has an inverse.^[4] Specifically, the inverse of an orthogonal transformation T is its adjoint T^\dagger, satisfying T^{-1} = T^\dagger, where the adjoint is defined with respect to the inner product.^[8] A key algebraic consequence is the closure under composition: if S and T are orthogonal transformations, then their composition S \circ T is also orthogonal, as it preserves the inner product by the successive application of the preservation property.^[9] This composition property, combined with invertibility and the existence of the identity transformation (which is orthogonal), establishes that the set of all orthogonal transformations on an n-dimensional real inner product space forms a group under composition, known as the orthogonal group O(n).^[1] Furthermore, the preservation of the inner product implies that orthogonal transformations maintain angles between vectors. For any vectors \mathbf{u} and \mathbf{v}, the cosine of the angle \theta between T(\mathbf{u}) and T(\mathbf{v}) equals the cosine of the angle between \mathbf{u} and \mathbf{v}, given by

\cos \theta = \frac{\langle T(\mathbf{u}), T(\mathbf{v}) \rangle}{\|T(\mathbf{u})\| \|T(\mathbf{v})\|} = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\|\mathbf{u}\| \|\mathbf{v}\|}

due to the inner product preservation and norm invariance.^[4] This angle preservation underscores the isometry nature of orthogonal transformations in the algebraic framework.^[8]

Matrix Representation

Orthogonal Matrices

An orthogonal transformation on an n-dimensional real vector space can be represented by an n \times n real matrix Q in a chosen orthonormal basis, where Q satisfies the condition Q^\top Q = I_n, with I_n denoting the n \times n identity matrix. This defining property ensures that the matrix Q is invertible with inverse equal to its transpose, Q^{-1} = Q^\top.^[10] The columns of such a matrix Q form an orthonormal basis for \mathbb{R}^n, meaning each column vector has unit length and the columns are pairwise orthogonal; the rows satisfy the same orthonormality condition. This orthonormality directly corresponds to the transformation preserving lengths and angles in the basis representation.^[10] The determinant of an orthogonal matrix Q is either +1 or -1, reflecting whether the transformation is orientation-preserving (a proper rotation) or orientation-reversing (including a reflection).^[10] One explicit method to construct an orthogonal matrix is via the Gram-Schmidt process, which orthogonalizes and normalizes a given set of linearly independent vectors to produce an orthonormal basis; the resulting basis vectors can then serve as the columns (or rows) of Q.^[11]

Eigenvalues and Singular Value Decomposition

The eigenvalues of a real orthogonal matrix lie on the unit circle in the complex plane, meaning their absolute values are all equal to 1.^[12] This property arises because if \lambda is an eigenvalue with eigenvector v, then Qv = \lambda v implies \|Qv\| = |\lambda| \|v\|, and since Q preserves norms, |\lambda| = 1. For real orthogonal matrices, the eigenvalues are either real, in which case they must be \pm 1, or they occur in complex conjugate pairs e^{i\theta} and e^{-i\theta} for some real \theta.^[12] The real eigenvalues \pm 1 correspond to invariant directions under the transformation, while the complex pairs reflect rotational components in the spectral decomposition. The singular value decomposition (SVD) provides a canonical factorization for any real matrix, and for a square orthogonal matrix Q, it simplifies significantly. Specifically, Q = U \Sigma V^T, where U and V are orthogonal matrices, and \Sigma is a diagonal matrix with all entries equal to 1 (the singular values of Q).^[13] This form underscores that orthogonal matrices have full rank and unit singular values, aligning with their norm-preserving nature; in fact, one possible SVD is Q = Q I I^T, highlighting the identity diagonal. A special case arises with proper orthogonal matrices, those with determinant 1, which form the special orthogonal group SO(n) and represent pure rotations without reflection.^[14] Their eigenvalues consist of \pm 1 (with even multiplicity for -1) and complex conjugate pairs on the unit circle, with 1 having multiplicity at least 1 in odd dimensions.^[15]

Geometric Aspects

Preservation of Inner Products

Orthogonal transformations, by definition, preserve the inner product in Euclidean space, meaning that for any linear map T: \mathbb{R}^n \to \mathbb{R}^n satisfying \langle T\mathbf{u}, T\mathbf{v} \rangle = \langle \mathbf{u}, \mathbf{v} \rangle for all vectors \mathbf{u}, \mathbf{v} \in \mathbb{R}^n, the geometric structure induced by the inner product remains unchanged.^[16] This preservation directly implies that norms are maintained, as \|T\mathbf{u}\| = \sqrt{\langle T\mathbf{u}, T\mathbf{u} \rangle} = \sqrt{\langle \mathbf{u}, \mathbf{u} \rangle} = \|\mathbf{u}\|, ensuring that lengths of vectors are invariant under T.^[17] Consequently, distances between points are also preserved, establishing orthogonal transformations as isometries of Euclidean space: d(T\mathbf{u}, T\mathbf{v}) = \|\mathbf{u} - \mathbf{v}\| = d(\mathbf{u}, \mathbf{v}).^[18] The angle \theta between two vectors \mathbf{u} and \mathbf{v} is defined via \cos \theta = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\|\mathbf{u}\| \|\mathbf{v}\|}, so the invariance of the inner product and norms ensures that angles are preserved under orthogonal transformations.^[16] This extends to orthogonality: two vectors \mathbf{u} and \mathbf{v} are perpendicular if \langle \mathbf{u}, \mathbf{v} \rangle = 0, and thus T\mathbf{u} \perp T\mathbf{v} if and only if \mathbf{u} \perp \mathbf{v}, maintaining perpendicular relationships in the space.^[17] In \mathbb{R}^n, this property implies that orthogonal transformations map the unit sphere \{\mathbf{x} \in \mathbb{R}^n : \|\mathbf{x}\| = 1\} to itself, as every point on the sphere has unit norm, which is preserved.^[19] In contrast to general linear transformations, which can distort shapes by scaling, shearing, or otherwise altering inner products, orthogonal transformations rigidly maintain the Euclidean metric without such deformations.^[16] This geometric fidelity distinguishes them as the linear maps that respect the full structure of distances and angles in Euclidean space.^[17]

Rotations and Reflections

Orthogonal transformations are broadly classified into two categories based on their determinant: proper orthogonal transformations with determinant 1, known as rotations, which preserve the orientation of the space; and improper orthogonal transformations with determinant -1, known as reflections, which reverse orientation. Rotations represent orientation-preserving isometries that maintain angles between vectors, making them essential for describing rigid body motions without flipping.^[15] In two dimensions, a rotation by an angle \theta around the origin is represented by the orthogonal matrix

\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix},

which has determinant 1 and preserves the counterclockwise sense of angles.^[15] This matrix rotates any vector (x, y) to (\cos \theta \cdot x - \sin \theta \cdot y, \sin \theta \cdot x + \cos \theta \cdot y), exemplifying how rotations act as circular shifts in the plane. Reflections, in contrast, are orientation-reversing orthogonal transformations, typically defined as reflections across a hyperplane, where the transformation fixes points in the hyperplane and negates the component along the normal direction.^[20] Such a reflection has eigenvalues of 1 with multiplicity n-1 in n-dimensional space (for the hyperplane directions) and a single eigenvalue of -1 corresponding to the unit normal vector to the hyperplane.^[20] A specific form used in numerical computations is the Householder reflection, given by the matrix I - 2 \mathbf{u} \mathbf{u}^T, where \mathbf{u} is a unit vector normal to the hyperplane and I is the identity matrix; this transformation is orthogonal with determinant -1 and is widely applied in algorithms like QR decomposition for its stability and efficiency.^[21] In three dimensions, rotations can be parameterized using Rodrigues' formula, which constructs the rotation matrix from an axis-angle representation, or via unit quaternions, which offer a compact, singularity-free alternative for composing multiple rotations.^[22]

Applications

In Physics and Engineering

In classical mechanics, the special orthogonal group SO(3) plays a central role in modeling the rotational dynamics of rigid bodies, where elements of SO(3) represent proper rotations that preserve orientation and distances in three-dimensional Euclidean space. These transformations are essential for deriving the equations of motion, such as Euler's rigid body equations, which describe how angular velocity evolves under torque-free conditions.^[23] The rotational invariance of physical laws under SO(3) symmetries leads to the conservation of angular momentum, as established by Noether's theorem, ensuring that the total angular momentum vector remains constant for isolated systems without external torques.^[24] In quantum mechanics, orthogonal transformations extend to their complex counterparts—unitary operators—which maintain the Hilbert space inner product and underpin both time evolution and symmetry operations. The time evolution operator U(t) = e^{-iHt/\hbar}, where H is the Hamiltonian, is unitary, guaranteeing the preservation of probability norms as quantum states propagate.^[25] Unitary representations of SO(3) describe spatial rotation symmetries, with angular momentum operators as generators, leading to conserved quantities like total angular momentum in systems invariant under rotations.^[26] Orthogonal transformations are vital in signal processing through orthogonal wavelet transforms, which decompose signals into a multiresolution basis of orthogonal functions for analysis and compression. The Haar wavelet transform, an early and computationally simple example, applies successive averaging and differencing operations to extract low-frequency approximations and high-frequency details, enabling efficient compression by thresholding insignificant coefficients while retaining perceptual quality in applications like image and audio processing.^[27] In robotics engineering, orthogonal transformations facilitate coordinate frame alignments to represent end-effector orientations relative to the base frame, often using rotation matrices from SO(3). Euler angles parametrize these orientations via sequential rotations about body-fixed or fixed axes, providing an intuitive three-parameter description for manipulator control and path planning. However, Euler angles exhibit singularities, such as gimbal lock, where alignment of rotation axes reduces the effective degrees of freedom, necessitating alternative representations like quaternions for robust operation.^[28]

In Statistics and Data Analysis

Orthogonal transformations play a central role in principal component analysis (PCA), a multivariate statistical technique used to reduce the dimensionality of data while preserving variance. In PCA, the covariance matrix of the centered data is orthogonally diagonalized to obtain the principal components, which are the eigenvectors of this matrix; these components form an orthogonal basis that transforms the original correlated variables into a new set of uncorrelated variables ordered by decreasing variance. This orthogonal diagonalization ensures that the transformed variables are independent in the sense of uncorrelatedness for multivariate normal distributions, facilitating the identification of the most informative directions in the data. The method was invented by Karl Pearson in 1901 and further developed by Harold Hotelling in 1933, who named it principal component analysis, as a way to analyze complexes of statistical variables.^[29]^[30] The Karhunen-Loève transform (KLT), closely related to PCA, provides an optimal orthogonal basis for representing stochastic processes or random vectors, particularly in data compression applications. By diagonalizing the autocorrelation matrix of the data ensemble, the KLT projects the signal onto its eigenvectors, which capture the maximum energy with the fewest components, minimizing the mean squared reconstruction error for a given number of retained basis functions. This transform is signal-dependent and achieves the theoretical lower bound on distortion for orthogonal expansions, making it ideal for compressing correlated data such as images or time series. The KLT was developed independently by Kari Karhunen in 1946 and Michel Loève in 1948, building on earlier work in spectral theory.^[31]^[32] In hypothesis testing within analysis of variance (ANOVA), orthogonal contrasts are employed to partition the total variation among group means into independent linear comparisons, ensuring that the tests for different hypotheses do not overlap in their use of the data. These contrasts, defined as linear combinations of group means with coefficients summing to zero and mutually orthogonal (i.e., their dot product is zero), allow for the decomposition of the sum of squares into non-overlapping components, maintaining the overall Type I error rate when performing multiple tests. For example, in a one-way ANOVA with multiple levels, orthogonal contrasts can test specific patterns like linear trends or comparisons between subsets of groups while preserving independence. This approach enhances the interpretability and power of post-hoc analyses in experimental designs.^[33] Orthogonal transformations also improve numerical stability in solving least squares problems through QR decomposition, where a matrix is factored into an orthogonal matrix Q and an upper triangular matrix R. In linear regression, applying QR decomposition to the design matrix avoids the ill-conditioning issues of the normal equations by preserving the Euclidean norm (since Q is orthogonal, ||Qx|| = ||x||), leading to more accurate solutions even for ill-conditioned systems. This method is particularly valuable in high-dimensional data analysis, where small perturbations in input can otherwise amplify errors exponentially. The stability benefits of QR-based least squares have been emphasized in numerical linear algebra literature since the 1960s.^[34]

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An orthogonal transformation is a linear transformation T:V->V which preserves a symmetric inner product. In particular, an orthogonal transformation ...
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A linear transformation A is orthogonal if and only if the column vectors of A form an orthonormal basis. Proof. Look at AT A = 1n. Each entry is a dot product ...Missing: definition algebra
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