Orthogonal matrices isolate the linear transformations of Euclidean space that preserve the metric structure: lengths, angles, and orthogonality. In linear algebra, a general invertible matrix may stretch space differently in different directions; an orthogonal matrix can rotate, reflect, or combine rotations and reflections without distortion. This makes orthogonal matrices the matrix-level meeting point of [inner product](/page/Inner%20Product), [linear map](/page/Linear%20Map), Euclidean norm, and change of orthonormal coordinates.

They are also a bridge between algebra and analysis. In multivariable calculus, a Jacobian matrix that is orthogonal at a point says that the derivative preserves lengths of tangent vectors at that point; by itself, this pointwise condition is not a global isometry statement. In [numerical analysis](/page/Numerical%20Analysis), orthogonal changes of variables are stable because they do not magnify Euclidean errors. In geometry, the [orthogonal group](/page/Orthogonal%20Group) packages the linear distance-preserving transformations of Euclidean space that fix the origin.

## Definition

The starting question is: which square matrices preserve the Euclidean inner product exactly? Since the inner product determines both length and angle, preserving it is the strongest finite-dimensional way to say that a linear transformation has no metric distortion. The transpose lets this condition be written as a single matrix identity.

[definition: Orthogonal Matrix] Let $n \in \mathbb{N}$. A matrix $Q \in \mathbb{R}^{n \times n}$ is an orthogonal matrix if \begin{align*} Q^\top Q = I_n. \end{align*} [/definition]

A single orthogonal matrix represents one rigid linear symmetry. Many arguments need to compose such symmetries, invert them, and compare them as a family. Naming the full collection records the fact that these matrices form a group of Euclidean symmetries.

[definition: Orthogonal Group] Let $n \in \mathbb{N}$. The orthogonal group $O(n)$ is the set \begin{align*} O(n) = \{Q \in \mathbb{R}^{n \times n} : Q^\top Q = I_n\}. \end{align*} [/definition]

Orientation is not recorded by the equation $Q^\top Q=I_n$. Rotations and reflections can both preserve distances, so the metric condition alone does not distinguish them. The determinant supplies the missing orientation information and leads to the orientation-preserving subgroup.

[definition: Special Orthogonal Group] Let $n \in \mathbb{N}$. The special orthogonal group $SO(n)$ is the set \begin{align*} SO(n) = \{Q \in O(n) : \det Q = 1\}. \end{align*} [/definition]

This subgroup isolates the determinant-$1$ part of $O(n)$. After the determinant constraint is established below, the complementary determinant-$-1$ part will account for transformations involving a reflection.

## Equivalent Characterisations

The matrix equation $Q^\top Q = I_n$ is compact, but it can hide the elementary column-by-column meaning. Each entry of $Q^\top Q$ is a dot product of two columns of $Q$. The next characterisation explains why orthogonal matrices are exactly the matrices whose columns form an orthonormal coordinate frame.

[quotetheorem:8283]

This column criterion is often the fastest way to test a concrete matrix: compute column lengths and mutual dot products, rather than multiplying out the whole matrix product. Its square-matrix hypothesis matters. Having orthonormal columns fills all of $\mathbb{R}^n$, so the matrix is automatically invertible; for rectangular matrices the same column calculation describes an isometric embedding, not an orthogonal matrix in $O(n)$. For geometric and analytic applications, however, the column test is still too coordinate-bound. What matters next is the metric effect of multiplying by $Q$: lengths, distances, and inner products should be unchanged.

A metric criterion is therefore needed because analytic estimates usually care about lengths and inner products rather than individual entries. The column criterion is computational, while applications to isometry require a coordinate-free interpretation. Orthogonality should mean that [Euclidean metric](/page/Euclidean%20Metric) data survives the transformation. The following theorem gives exactly that interpretation.

[quotetheorem:439]

Coordinate changes are where the metric characterisation becomes indispensable. An arbitrary [change of basis](/page/Change%20Of%20Basis) may turn a unit vector into a coordinate vector of a different Euclidean length, so matrix estimates can become contaminated by the chosen basis. When both bases are orthonormal, the change-of-basis matrix should preserve the coordinate inner product, and the next theorem records that orthonormal coordinate changes are exactly the safe ones.

[quotetheorem:8598]

This theorem is the coordinate bookkeeping behind many later uses of orthogonal matrices. To see the two basic geometric behaviours before moving to general properties, we start with the planar matrices that rotate or reflect the standard orthonormal frame.

## Standard Examples