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Orthogonal
Matrices

Discover matrices that preserve lengths and angles: the building blocks of rotations, reflections, and dimensionality reduction.

Linear Algebra

What Is an Orthogonal Matrix?

A square matrix \( Q \) is called orthogonal if its transpose equals its inverse:

This single condition encodes a powerful geometric idea: when you multiply a vector by \( Q \), the length of the vector and the angle between any two vectors are unchanged.

The Column Perspective

The condition \( Q^T Q = I \) tells us something elegant about the columns of \( Q \). Let \( \mathbf{q}_1, \mathbf{q}_2, \ldots, \mathbf{q}_n \) be the columns:

Orthonormal
Every pair of distinct columns is orthogonal: \( \mathbf{q}_i^T \mathbf{q}_j = 0 \) for \( i \neq j \).
Unit Length
Each column has unit norm: \( \mathbf{q}_i^T \mathbf{q}_i = 1 \) for every \( i \).

Geometric Meaning

Before \( Q \) u v \( \theta \) → Multiply by Q After \( Q \) Qu Qv \( \theta \) |u|, |v| unchanged Angle θ preserved

Core Properties of Orthogonal Matrices

1
Inverse equals transpose: \( Q^{-1} = Q^T \).
2
Length preservation: \( \|Q\mathbf{x}\| = \|\mathbf{x}\| \) for all \( \mathbf{x} \).
3
Determinant: \( \det(Q) = +1 \) (rotation) or \( \det(Q) = -1 \) (reflection).
4
Eigenvalues: Every eigenvalue \( \lambda \) satisfies \( |\lambda| = 1 \).
5
Closure: The product of two orthogonal matrices is orthogonal.

Why \(\det(Q) = \pm 1\)?

Start from the defining equation and take the determinant of both sides:

\(\det = +1\)
Proper rotation (special orthogonal). Orientation is preserved.
\(\det = -1\)
Improper rotation (includes a reflection). Orientation is reversed.

Rotation vs Reflection

Rotation (det = +1) x Qx \( \theta \) Orientation preserved Reflection (det = −1) mirror line x Qx Orientation reversed

Key Distinctions

Property Orthogonal Matrix Unitary Matrix
Field Real (\( \mathbb{R} \)) Complex (\( \mathbb{C} \))
Defining equation \( Q^T Q = I \) \( U^* U = I \)
Conjugate? No (transpose only) Yes (conjugate transpose)
Determinant \( \pm 1 \) \( |\det(U)| = 1 \)
Eigenvalues \( |\lambda| = 1 \) \( |\lambda| = 1 \)

How to Verify Orthogonality

1
Check square: The matrix must be \( n \times n \).
2
Compute column norms: Verify \( \|\mathbf{q}_i\| = 1 \) for each column.
3
Check orthogonality: Verify \( \mathbf{q}_i^T \mathbf{q}_j = 0 \) for all \( i \neq j \).
4
Shortcut: Compute \( Q^T Q \) and confirm it equals \( I \).

Example 1: A 2×2 Rotation Matrix

Consider the rotation matrix for angle \( \theta = 60^\circ \):

Verify column norms and orthogonality:

Example 2: A Reflection Matrix

Reflection across the line \( y = x \):

1
Columns: \( \mathbf{r}_1 = \begin{pmatrix}0\\1\end{pmatrix} \), \( \mathbf{r}_2 = \begin{pmatrix}1\\0\end{pmatrix} \). Both unit length ✓
2
Dot product: \( \mathbf{r}_1^T \mathbf{r}_2 = 0\cdot 1 + 1\cdot 0 = 0 \). Orthogonal ✓
3
Determinant: \( \det(R) = 0\cdot 0 - 1\cdot 1 = -1 \). Reflection ✓

Example 3: A 3×3 Orthogonal Matrix

Is this matrix orthogonal?

Check \( Q^T Q \): Each column has norm \( \frac{1}{3}\sqrt{4+4+1} = \frac{1}{3}\cdot 3 = 1 \) ✓

Why Lengths Are Preserved

The proof is elegant and short:

Taking square roots: \( \|Q\mathbf{x}\| = \|\mathbf{x}\| \). The same argument shows angles are preserved, since inner products are preserved:

Products Stay Orthogonal

If \( Q_1 \) and \( Q_2 \) are both orthogonal, then so is their product \( Q_1 Q_2 \):

Group Structure
Orthogonal matrices form the orthogonal group \( O(n) \) under multiplication. It is closed, associative, has identity \( I \), and each element has inverse \( Q^T \).
Special Orthogonal
Restricting to \( \det = +1 \) gives the special orthogonal group \( SO(n) \): the group of pure rotations.

Common Mistakes

✕ Confusing "orthogonal" with "orthonormal"
A matrix with orthogonal columns is NOT orthogonal unless each column also has unit length. Scale matters.
✕ Forgetting squareness
A rectangular matrix can have orthonormal columns, but it is not called "orthogonal." The definition requires \( n \times n \).
✕ Assuming det = +1
Reflections are orthogonal too. Don't assume every orthogonal matrix is a rotation: always check the determinant.

Summary of Key Facts

Property Statement
Definition \( Q^T Q = I \)
Inverse \( Q^{-1} = Q^T \)
Columns Form an orthonormal basis
Determinant \( \pm 1 \)
Eigenvalues \( |\lambda| = 1 \)
Preserves Lengths, angles, inner products
Group \( O(n) \); rotations form \( SO(n) \)

Question 1 of 5

If \( Q \) is an orthogonal matrix, which of the following must be true?

Question 2 of 5

What is the determinant of a 2D rotation matrix \( R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \)?

Question 3 of 5

Let \( Q \) be orthogonal and \( \mathbf{x} \in \mathbb{R}^n \). What is \( \|Q\mathbf{x}\| \)?

Question 4 of 5

Which matrix is NOT orthogonal?

Question 5 of 5

If \( Q_1 \) and \( Q_2 \) are orthogonal matrices, what can you say about \( Q_1 Q_2 \)?