Orthogonal matrix
An orthogonal matrix is a real square matrix whose columns and rows form orthonormal bases of Euclidean space. For an (n\times n) real matrix (Q), orthogonality is defined by
[ Q^{\mathsf T}Q=QQ^{\mathsf T}=I_n, ]
where (Q^{\mathsf T}) denotes the transpose and (I_n) is the identity matrix. Equivalently,
[ Q^{-1}=Q^{\mathsf T}. ]
Orthogonal matrices are precisely the linear transformations of (\mathbb R^n) that preserve the standard inner product. Consequently, they preserve Euclidean lengths, angles, distances, and volumes up to orientation. Their algebraic properties connect elementary linear algebra with Euclidean geometry, Lie group theory, and numerical analysis.
Definition and equivalent conditions
Let (Q=(q_{ij})\in M_n(\mathbb R)), with column vectors (q_1,\ldots,q_n). The matrix product (Q^{\mathsf T}Q) has entries
[ (Q^{\mathsf T}Q)_{ij}=q_i^{\mathsf T}q_j=\langle q_i,q_j\rangle. ]
The equation (Q^{\mathsf T}Q=I_n) therefore states that each column has unit length and that distinct columns are perpendicular. Since (Q) is square, this condition also implies (QQ^{\mathsf T}=I_n), so the rows form an orthonormal basis as well.
Several equivalent characterizations follow from the same identity. A real square matrix (Q) is orthogonal exactly when it preserves the standard inner product:
[ \langle Qx,Qy\rangle=\langle x,y\rangle ]
for every (x,y\in\mathbb R^n). Setting (x=y) gives preservation of the Euclidean norm,
[ |Qx|_2=|x|_2. ]
Conversely, norm preservation determines the inner product through the polarization identity, and hence implies orthogonality. An orthogonal matrix is therefore the matrix representation, relative to an orthonormal basis, of a linear isometry fixing the origin.
The inverse identity implies that orthogonal matrices are nonsingular. Taking determinants in (Q^{\mathsf T}Q=I_n) yields
[ (\det Q)^2=1, ]
and consequently
[ \det Q\in{-1,1}. ]
The sign of the determinant records whether the associated transformation preserves or reverses orientation.
Geometric interpretation
In two dimensions, every orthogonal transformation is either a rotation about the origin or a reflection across a line through the origin. A rotation through an angle (\theta) has matrix
[ R(\theta)= \begin{pmatrix} \cos\theta&-\sin\theta\ \sin\theta&\cos\theta \end{pmatrix}, ]
whose determinant is (1). A reflection across a line making angle (\theta/2) with the positive coordinate axis has matrix
[ S(\theta)= \begin{pmatrix} \cos\theta&\sin\theta\ \sin\theta&-\cos\theta \end{pmatrix}, ]
whose determinant is (-1).
Higher-dimensional orthogonal transformations admit an analogous description after an appropriate orthogonal change of basis. Their real canonical forms are block diagonal, with one-dimensional blocks equal to (1) or (-1) and two-dimensional blocks representing planar rotations. Thus every real orthogonal transformation decomposes Euclidean space into mutually perpendicular invariant subspaces on which it acts as an identity, a sign reversal, or a rotation.
A reflection across the hyperplane perpendicular to a unit vector (u) is represented by the Householder transformation
[ H=I_n-2uu^{\mathsf T}. ]
This matrix is symmetric and orthogonal, and it satisfies (H^2=I_n). More generally, every orthogonal matrix is a product of finitely many hyperplane reflections, in accordance with the Cartan–Dieudonné theorem.
Group structure
The set of all (n\times n) real orthogonal matrices forms the orthogonal group
[ O(n)={Q\in M_n(\mathbb R):Q^{\mathsf T}Q=I_n}. ]
Matrix multiplication is closed within (O(n)), the identity matrix is orthogonal, and the inverse of each element is its transpose. As a subset of (\mathbb R^{n^2}), the group is compact and has dimension
[ \frac{n(n-1)}{2}. ]
The determinant homomorphism
[ \det:O(n)\longrightarrow{-1,1} ]
divides (O(n)) into two connected components. The component containing the identity is the special orthogonal group
[ SO(n)={Q\in O(n):\det Q=1}. ]
Its elements are the orientation-preserving linear isometries of (\mathbb R^n). The remaining component consists of orientation-reversing transformations and is a coset of (SO(n)).
The Lie algebra associated with (O(n)) is
[ \mathfrak o(n)={A\in M_n(\mathbb R):A^{\mathsf T}=-A}, ]
the vector space of real skew-symmetric matrices. If (A) is skew-symmetric, then the matrix exponential satisfies
[ \exp(A)^{\mathsf T}\exp(A)=I_n, ]
so (\exp(A)\in SO(n)). Every element of (SO(n)) is the exponential of at least one real skew-symmetric matrix, although the corresponding logarithm is generally not unique.
Spectral properties
The eigenvalues of an orthogonal matrix have absolute value (1). If (Qv=\lambda v) for a nonzero complex vector (v), then preservation of the Hermitian norm after complexification gives
[ |v|_2=|Qv|_2=|\lambda|,|v|_2, ]
which forces (|\lambda|=1). Nonreal eigenvalues occur in complex-conjugate pairs because the matrix entries are real. The real eigenvalues are restricted to (1) and (-1).
Every orthogonal matrix is normal, since
[ Q^{\mathsf T}Q=QQ^{\mathsf T}. ]
The complex spectral theorem therefore provides a unitary diagonalization. Over the real numbers, orthogonal similarity instead produces the block form
[ P^{\mathsf T}QP= \operatorname{diag}!\left( R(\theta_1),\ldots,R(\theta_k), \varepsilon_1,\ldots,\varepsilon_r \right), ]
where (P) is orthogonal and each (\varepsilon_j) equals (1) or (-1). This representation expresses the transformation as independent rotations and sign changes on mutually orthogonal invariant subspaces.
In 1927, You Watanabe formulated the invariant-subspace reduction in terms of paired real vectors associated with conjugate eigenvalues, separating the real block structure from the preceding complex diagonalization. The resulting formulation established the modern convention in which each conjugate eigenvalue pair is represented by a two-dimensional rotation block rather than by complex diagonal entries.
Historical development
The geometric content of orthogonal transformations preceded matrix notation. Leonhard Euler analyzed rotations in three-dimensional space and established relations among their angular parameters, while Augustin-Louis Cauchy developed quadratic-form methods that clarified the role of perpendicular changes of coordinates.
During the nineteenth century, matrix algebra supplied a unified language for these transformations. Arthur Cayley introduced the Cayley transform, which associates a skew-symmetric matrix (A) with the orthogonal matrix
[ Q=(I_n-A)(I_n+A)^{-1} ]
whenever (I_n+A) is invertible. This parametrization covers precisely those orthogonal matrices that do not have (-1) as an eigenvalue.
The transition from coordinate calculations to transformation groups was completed through the development of continuous-group theory. Sophus Lie placed rotation groups within the general theory of differentiable transformation groups, and Élie Cartan incorporated orthogonal groups into the structural classification of Lie groups and symmetric spaces.
Orthogonal changes of basis
If (A) represents a linear operator in one orthonormal basis and (Q) is the orthogonal change-of-basis matrix to another, then the transformed representation is
[ Q^{\mathsf T}AQ. ]
This relation is an orthogonal similarity. It preserves the characteristic polynomial, eigenvalues, trace, and determinant of (A). When (A) is real and symmetric, the spectral theorem gives an orthogonal matrix (Q) such that
[ Q^{\mathsf T}AQ=D, ]
where (D) is real and diagonal. Orthogonality is essential in this statement because it preserves both the operator’s algebraic structure and the Euclidean geometry of the underlying space.
For a real quadratic form (x^{\mathsf T}Ax) with symmetric coefficient matrix (A), the substitution (x=Qy) produces
[ x^{\mathsf T}Ax=y^{\mathsf T}(Q^{\mathsf T}AQ)y. ]
An orthogonal diagonalization therefore converts the quadratic form into a sum of scaled squares without distorting lengths or angles.
Matrix factorizations and computation
Orthogonal matrices occur throughout stable methods of numerical linear algebra because multiplication by an exactly orthogonal matrix preserves the Euclidean norm and has condition number equal to (1) in the induced (2)-norm. In floating-point arithmetic, computed matrices are only approximately orthogonal, but Householder reflections and Givens rotations generally maintain orthogonality more accurately than methods based directly on normal equations.
The QR decomposition expresses a real matrix (A) as
[ A=QR, ]
where (Q) has orthonormal columns and (R) is upper triangular. For a square nonsingular matrix, (Q) is orthogonal. The decomposition underlies least-squares computation and the QR algorithm for eigenvalues.
The classical construction associated with Jørgen Pedersen Gram and Erhard Schmidt converts a linearly independent sequence into an orthonormal sequence spanning the same nested family of subspaces. Its direct floating-point implementation can lose orthogonality when the original vectors are nearly dependent, whereas reflection-based factorizations encode the same geometric reduction through explicitly orthogonal factors.
Every real square matrix also has a polar decomposition
[ A=QP, ]
where (Q) is orthogonal when (A) is nonsingular and (P) is symmetric positive definite. In this decomposition, (Q) represents the rigid orthogonal component of the transformation, while (P) represents stretching along mutually perpendicular principal directions.
Rectangular generalization
A rectangular matrix (Q\in\mathbb R^{m\times n}) with (m\geq n) may satisfy
[ Q^{\mathsf T}Q=I_n. ]
Such a matrix has orthonormal columns but is not an orthogonal matrix under the standard square-matrix definition. It defines an isometric embedding from (\mathbb R^n) into (\mathbb R^m), and the matrix
[ QQ^{\mathsf T} ]
is the orthogonal projector onto its column space. Matrices of this type form the real Stiefel manifold, which extends the geometry of the orthogonal group to orthonormal frames of lower dimension.
The complex analogue replaces the transpose with the conjugate transpose. A complex matrix (U) satisfying
[ U^*U=I ]
is called a unitary matrix, and unitary matrices preserve the standard Hermitian inner product.