Orthogonal Matrices
Preserve length: QᵀQ = I.
Key idea
Preserve length: QᵀQ = I.
In depth
Q is orthogonal if QᵀQ = I, meaning its columns are orthonormal. Such matrices preserve lengths and angles (rotations and reflections), and Q⁻¹ = Qᵀ.
Worked example
The rotation matrix [cos θ −sin θ; sin θ cos θ] is orthogonal.
Formula
Real-world uses
- Rotating 3D objects without distortion
- Stable computations in numerical software
Common mistake
Only checking that columns are orthogonal, but not unit length.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Orthogonal Matrices?
Preserve length: QᵀQ = I. It is a topic at the expert level in Linear Algebra.
What is the formula for Orthogonal Matrices?
The key formula is QᵀQ = I.
Where is Orthogonal Matrices used in real life?
Real-world examples: Rotating 3D objects without distortion; Stable computations in numerical software.
What should I learn before Orthogonal Matrices?
Make sure you are comfortable with Orthogonal Projections, the previous topic in Linear Algebra.
What is a common mistake with Orthogonal Matrices?
Only checking that columns are orthogonal, but not unit length.
How can I practise Orthogonal Matrices?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Orthogonal Matrices?
Move on to Eigenvalues and Eigenvectors next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.