# 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.