Chapter 21 Quiz — Orthogonal Matrices and Rotations
Twelve conceptual checks. Try each before opening the answer. One-line explanations follow every answer.
Q1. Which single matrix equation defines a real orthogonal matrix $Q$?
Answer
$Q^{\mathsf{T}}Q = I$ — equivalently $Q^{-1} = Q^{\mathsf{T}}$. *The transpose is the inverse; this is the whole definition.*Q2. "Orthogonal matrix" requires the columns to be merely perpendicular. True or false?
Answer
**False.** The columns must be **orthonormal** — perpendicular *and* unit length. *A matrix with perpendicular but non-unit columns, like $\mathrm{diag}(2,3)$, is not orthogonal.*Q3. What two geometric quantities does every orthogonal matrix preserve?
Answer
**Lengths and angles** (and therefore areas/volumes too). *Orthogonal matrices are the isometries — the rigid motions about a fixed origin.*Q4. The determinant of an orthogonal matrix can be any number with absolute value 1. True or false?
Answer
**False** for a *real* orthogonal matrix: its determinant is exactly $+1$ or $-1$. *Only a complex **unitary** matrix can have a determinant anywhere on the unit circle, e.g. $\det = i$.*Q5. A $2\times2$ matrix has $\det = -1$ and satisfies $Q^{\mathsf{T}}Q = I$. Is it a rotation or a reflection?
Answer
A **reflection**. *Determinant $-1$ means orientation is reversed; rotations have $\det = +1$.*Q6. You compose a rotation with a reflection. What is the determinant of the result, and which type of transformation is it?
Answer
$\det = (+1)(-1) = -1$, so the result is a **reflection**. *Determinants multiply under composition.*Q7. Why is solving $Q\mathbf{x} = \mathbf{b}$ so cheap when $Q$ is orthogonal?
Answer
Because $Q^{-1} = Q^{\mathsf{T}}$, so $\mathbf{x} = Q^{\mathsf{T}}\mathbf{b}$ — just a transpose-times-vector, with **no** Gaussian elimination, determinant, or division. *Orthogonal matrices are the cheapest to invert and are never near-singular.*Q8. How many real numbers (degrees of freedom) does a general 3D rotation require, and why?
Answer
**Three**: two to pick the axis (a direction on the sphere) and one for the angle of spin. *Equivalently, $9$ matrix entries minus $6$ orthonormality constraints $= 3$.*Q9. What is a Householder reflection, and what is its determinant?
Answer
$H = I - 2\mathbf{u}\mathbf{u}^{\mathsf{T}}$ for a unit vector $\mathbf{u}$; it reflects across the hyperplane perpendicular to $\mathbf{u}$, and $\det(H) = -1$. *It is symmetric and its own inverse ($H^2 = I$).*Q10. What is the complex analogue of an orthogonal matrix, and what equation defines it?
Answer
A **unitary** matrix $U$, defined by $U^{*}U = I$, where $U^{*}$ is the conjugate (Hermitian) transpose. *Every real orthogonal matrix is unitary; the conjugate is what makes complex "length" come out real and non-negative.*Q11. Why must every quantum logic gate be a unitary matrix?
Answer
A gate sends a qubit $\mathbf{z}\mapsto U\mathbf{z}$, and the total probability $\lVert\mathbf{z}\rVert^2$ must stay equal to 1. *Only length-preserving (unitary) maps keep probabilities summing to 1; this also makes every gate reversible, since $U^{-1} = U^{*}$.*Q12. A friend claims the shear $\begin{psmallmatrix}1 & 7\\ 0 & 1\end{psmallmatrix}$ is orthogonal "because its determinant is 1." What is wrong?