Orthogonal Matrix — Properties, Examples

TL;DR

An orthogonal matrix is a square matrix A whose transpose equals its inverse — equivalently, ( A^TA = AA^T = I ). This article covers the formal definition, the property list, three worked examples at Quick/Standard/Stretch tiers, and the role of orthogonal matrices in 3D rotations, computer graphics, and quantum mechanics.

A Matrix That Rotates Without Distorting

Most matrices stretch, squash, and shear the vectors they act on. A small set of square matrices — the orthogonal ones — does something different: they rotate (or reflect) vectors without changing any vector's length.

What an Orthogonal Matrix Is

A square matrix A of size ( n \times n ) is orthogonal if its transpose equals its inverse: [ A^TA = AA^T = I_n. ] Equivalently — and this is the property most often used in practice — the columns of A form an orthonormal set: each column has length 1, and any two distinct columns are perpendicular (their dot product is 0). The rows form the same kind of set.

Quick facts:

• Definition: ( A^TA = AA^T = I ), equivalently ( A^{-1} = A^T. )
• Determinant: ( det(A) = \pm 1 ).
• Columns and rows: orthonormal — unit length, pairwise perpendicular.
• Length-preserving: ( |Av| = |v| ) for every vector v.
• Eigenvalues: complex of modulus 1 — ( |\lambda| = 1 ) for every eigenvalue.
• Closed under multiplication: the product of two orthogonal matrices is orthogonal.

The Four Key Properties of Orthogonal Matrix

1. The transpose is the inverse

[ A^{-1} = A^T. ] Computing the inverse of an orthogonal matrix is free — just transpose.

2. Lengths are preserved

[ |Av|^2 = (Av)^T(Av) = v^T A^TA v = v^TIv = |v|^2. ] Every vector keeps its length after the transformation.

3. Angles are preserved

[ (Au)\cdot(Av) = u^TA^TA v = u^Tv = u\cdot v. ] Every angle between vectors keeps its measure.

4. The determinant is ±1

[ det(A^TA) = det(I) = 1, ] and ( det(A^T) = det(A) ), so ( det(A)^2 = 1 ), hence ( det(A) = \pm 1. )

Worked Examples of Orthogonal Matrix

Quick. Verify that ( A = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} ) is orthogonal.

[ A^T = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}, \quad A^TA = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} = I. ] Final answer: A is orthogonal.

Standard (Wrong Path First — A Common Slip Worth Walking Through). Check whether ( B = \begin{pmatrix} 2 & 0 \ 0 & 2 \end{pmatrix} ) is orthogonal.

[ B^TB = \begin{pmatrix} 2 & 0 \ 0 & 2 \end{pmatrix} \begin{pmatrix} 2 & 0 \ 0 & 2 \end{pmatrix} = \begin{pmatrix} 4 & 0 \ 0 & 4 \end{pmatrix} \neq I. ] Final answer: B is not orthogonal.

Stretch. Show that the 2D rotation matrix ( R = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix} ) is orthogonal for every angle ( \theta ).

Final answer: the rotation matrix is orthogonal for every ( \theta ).

Where Orthogonal Matrices Show Up — From Graphics to Quantum Mechanics

• Computer graphics: Every 3D rotation of a video-game character is a 3×3 orthogonal matrix.
• Robotics: A robot arm's pose is a sequence of orthogonal-matrix rotations.
• Quantum mechanics: The algebraic name for symmetries of physical systems.

Where Things Go Sideways — Common Mistakes of Orthogonal Matrix

1. Confusing orthogonal columns with orthonormal columns.
2. Skipping the ( A^TA = I ) check.
3. Forgetting the determinant sign.
4. Mixing up orthogonal matrices with symmetric matrices.

The Mathematicians Who Shaped Matrix Algebra

• Arthur Cayley (1821–1895, England) introduced modern matrix theory.
• James Joseph Sylvester (1814–1897, England) coined the term "matrix."
• Camille Jordan (1838–1922, France) studied the orthogonal group.

Conclusion

An orthogonal matrix is a square matrix A where ( A^TA = I ), equivalently ( A^{-1} = A^T. ) The columns (and rows) form an orthonormal set — unit length and pairwise perpendicular. Orthogonal matrices preserve both lengths and angles — they are the algebraic name for rigid motion.