Orthogonal Matrix

An orthogonal matrix is a square matrix with real entries whose columns, and equivalently whose rows, form an orthonormal basis. Its defining equation is $Q^{T}Q=I$, where $Q^{T}$ denotes its transpose and $I$ the identity matrix. In linear algebra, orthogonal matrices represent transformations that preserve lengths and angles, making them fundamental to geometry, changes of coordinates, and numerical computation. Despite the name, their columns must be both mutually perpendicular and of unit length, not merely perpendicular. (graphsearch.epfl.ch)

Definition and equivalent conditions

For $Q\in\mathbb{R}^{n\times n}$, orthogonality means

Here $Q^{T}$ is the matrix transpose, and $I_n$ is the identity matrix. Equivalently, $Q$ is invertible and its inverse satisfies $Q^{-1}=Q^{T}$. Because $Q$ is square, either product equation implies the other. (math.mit.edu)

Writing its columns as $q_1,\ldots,q_n$, the defining condition becomes

Thus the columns constitute a basis of the vector space $\mathbb{R}^{n}$. The same statement holds for the rows. In particular, orthogonality describes an entire coordinate frame rather than just a collection of perpendicular directions. (graphsearch.epfl.ch)

Geometric interpretation

An orthogonal matrix defines a linear map $x\mapsto Qx$ preserving the standard inner product:

Consequently, it preserves vector lengths, angles between nonzero vectors, and Euclidean distances:

It fixes the origin, so a translation alone is not represented by an orthogonal matrix. A distance-preserving transformation that also translates points instead has the affine form $x\mapsto Qx+b$. (web.ma.utexas.edu)

A basic example is the planar rotation matrix

It rotates column vectors counterclockwise through angle $\theta$. The matrix $\operatorname{diag}(1,-1)$, by contrast, reflects the plane across the horizontal axis. Both preserve lengths, but they affect orientation differently. (math.mit.edu)

Determinants and group structure

Taking the determinant of $Q^{T}Q=I$ gives

so every orthogonal matrix has determinant $+1$ or $-1$. The first case preserves orientation; the second reverses it. Both preserve the magnitude of volume. Determinant $+1$ alone, however, does not establish orthogonality: a matrix may preserve volume while stretching some directions and contracting others. (math.mit.edu)

Products and inverses of orthogonal matrices are orthogonal. Together with the identity, all $n\times n$ orthogonal matrices therefore form the orthogonal group $O(n)$, an important object in group theory. Its determinant-$+1$ subgroup is the special orthogonal group $SO(n)$. In two dimensions, this subgroup consists of planar rotations; the determinant-$-1$ matrices are reflections across lines through the origin. (math.mit.edu)

Eigenvalues and related matrix classes

The eigenvalues of an orthogonal matrix have absolute value one. Although the matrix has real entries, its eigenvalues may be complex numbers. If $Qv=\lambda v$ for a nonzero complex vector, preservation of the complex Euclidean norm gives

Hence $|\lambda|=1$. Real eigenvalues can only be $+1$ or $-1$, while nonreal eigenvalues occur in conjugate pairs. A planar rotation has eigenvalues $e^{i\theta}$ and $e^{-i\theta}$, so it generally has no real eigenvectors. (ocw.mit.edu)

The corresponding length-preserving class over complex coordinates is the unitary matrix, defined by $U^{*}U=I$, using conjugate transpose rather than ordinary transpose. Every real orthogonal matrix is unitary when regarded as a complex matrix. (netlib.org)

Construction and numerical uses

The Gram–Schmidt process constructs an orthonormal basis from linearly independent vectors. Arranging a complete such basis as columns produces an orthogonal matrix. Orthogonal factors also appear in QR decomposition, which expresses a real matrix as $A=QR$, with $R$ triangular or trapezoidal. A reduced QR factorization may instead use a rectangular $Q$ with orthonormal columns. (math.mit.edu)

Computational implementations commonly build orthogonal factors from Householder transformations or Givens rotations. A Householder reflection has the form

It reflects across the hyperplane perpendicular to $v$. These elementary transformations can introduce zeros into a matrix while retaining its Euclidean geometry. (netlib.org)

Orthogonal transformations are central to numerical linear algebra, including least-squares problems and singular value decomposition. Their norm preservation supports numerical stability because applying an exact orthogonal transformation does not amplify an existing error measured in the Euclidean norm. This does not eliminate rounding errors introduced during computation. (netlib.org)

Rectangular matrices and projections

For an $m\times n$ matrix $Q$ with $m>n$, the condition $Q^{T}Q=I_n$ describes orthonormal columns, but $QQ^{T}\ne I_m$. Instead, $QQ^{T}$ represents the orthogonal projection onto their span. Thus reduced orthogonal factors must be distinguished from square orthogonal matrices: they preserve lengths when mapping $\mathbb{R}^{n}$ into $\mathbb{R}^{m}$, but do not provide an invertible transformation of the whole larger space. (netlib.sandia.gov)