Brownian Bridge™

1. The SVD of $A \in \mathbb{R}^{m \times n}$ (with $m \geq n$ ) is $A = U\Sigma V^\top$ . Which statement correctly describes the shapes and properties of the three factors in the **thin** (economy) SVD?

U ∈ ℝ^(m×n) has orthonormal columns, Σ = diag(σ₁,…,σₙ) ∈ ℝ^(n×n), V ∈ ℝ^(n×n) is orthogonalU ∈ ℝ^(m×m) is orthogonal, Σ ∈ ℝ^(m×n) is rectangular diagonal, V ∈ ℝ^(n×n) is orthogonalU ∈ ℝ^(m×n) has orthonormal columns, Σ ∈ ℝ^(m×n) is rectangular diagonal, V ∈ ℝ^(n×n) is orthogonalU ∈ ℝ^(m×m) is orthogonal, Σ = diag(σ₁,…,σₙ) ∈ ℝ^(n×n), V ∈ ℝ^(n×n) is orthogonal

2. Let $A = \begin{pmatrix} 3 & 0 \\ 0 & 0 \\ 0 & 2 \end{pmatrix} \in \mathbb{R}^{3 \times 2}$ . What are the singular values of $A$ , and what is $\kappa_2(A)$ ?

σ₁ = 3, σ₂ = 2, κ₂ = 3/2σ₁ = 3, σ₂ = 2, κ₂ = 9/4σ₁ = 9, σ₂ = 4, κ₂ = 9/4σ₁ = 3, σ₂ = 0, κ₂ = ∞

3. The Moore-Penrose pseudoinverse of $A \in \mathbb{R}^{m \times n}$ (with $m > n$ and full column rank $r = n$ ) satisfies four conditions. Which of the following is the correct closed form and which condition does it satisfy that is **not** satisfied by an arbitrary left inverse?

A⁺ = VΣ⁻¹Uₙᵀ; it is the unique left inverse satisfying (A⁺A)ᵀ = A⁺AA⁺ = (AᵀA)Aᵀ; it minimises the residual ‖Ax − b‖₂A⁺ = UₙΣ⁻¹Vᵀ; it is the left inverse with largest spectral normA⁺ = VΣ⁻¹Uₙᵀ; it always returns the maximum-norm solution x

4. A risk quant solves $Ax = b$ where $A \in \mathbb{R}^{50 \times 10}$ is a sensitivity matrix. The singular values of $A$ are $\sigma_1 = 100$ , $\sigma_2 = 50$ , …, $\sigma_9 = 1$ , $\sigma_{10} = 10^{-5}$ . The market quotes $b$ have a relative error of $10^{-4}$ (bid-ask spread). What is the worst-case relative error in the recovered parameters $x$ ?

κ₂(A) × 10⁻⁴ = 100/10⁻⁵ × 10⁻⁴ = 10³σ₁ × 10⁻⁴ = 100 × 10⁻⁴ = 10⁻²σ₁₀ × 10⁻⁴ = 10⁻⁵ × 10⁻⁴ = 10⁻⁹1/(σ₁₀ × 10⁻⁴) = 10⁹

5. For $A \in \mathbb{R}^{m \times n}$ with thin SVD $A = U_r \Sigma_r V_r^\top$ (rank $r$ ), the rank- $k$ Eckart-Young approximation $A_k = \sum_{i=1}^k \sigma_i u_i v_i^\top$ satisfies two error formulas. Which pair is correct?

‖A − Aₖ‖₂ = σₖ₊₁ and ‖A − Aₖ‖_F = √(Σᵢ₌ₖ₊₁ʳ σᵢ²)‖A − Aₖ‖₂ = σₖ₊₁ and ‖A − Aₖ‖_F = Σᵢ₌ₖ₊₁ʳ σᵢ‖A − Aₖ‖₂ = σₖ² and ‖A − Aₖ‖_F = √(Σᵢ₌ₖ₊₁ʳ σᵢ²)‖A − Aₖ‖₂ = √(Σᵢ₌ₖ₊₁ʳ σᵢ²) and ‖A − Aₖ‖_F = σₖ₊₁

6. A portfolio risk system computes the covariance matrix $\hat{\Sigma} = X^\top X / T$ from $T = 50$ daily returns on $n = 200$ assets. A risk quant wants to invert $\hat{\Sigma}$ to compute minimum-variance weights. Which statement is correct?

Σ̂ is rank-deficient (rank ≤ 50), so it is not invertible; use the pseudoinverse or add a regularisation term λIΣ̂ is always invertible because it is positive semi-definiteΣ̂ has condition number exactly T/n = 50/200 = 0.25, so it is well-conditionedThe pseudoinverse Σ̂⁺ gives the minimum-variance portfolio that uses all 200 assets

7. You apply Tikhonov regularisation to an ill-conditioned calibration problem: $x_\lambda = \sum_{i=1}^n \frac{\sigma_i}{\sigma_i^2 + \lambda} (u_i^\top b)\, v_i$ . As $\lambda \to \infty$ , what happens to $x_\lambda$ , and what does this mean in financial terms?

x_λ → 0; all parameters are shrunk to zero regardless of the data — equivalent to a prior centred at the origin with infinite precisionx_λ → A⁺b; Tikhonov converges to the pseudoinverse solutionx_λ → the maximum-likelihood estimate; regularisation has no effect as λ → ∞x_λ → x_true; Tikhonov always recovers the true parameters for large λ

8. A senior quant asks why the implied vol surface calibration on Monday and Tuesday gives SABR parameters that look completely different, even though the option prices barely moved. The sensitivity matrix $\partial F/\partial\theta$ has singular values $[8.2, 3.1, 0.9, 0.04, 0.04]$ . What is the correct diagnosis and remediation?

σ₄ ≈ σ₅ ≈ 0.04: the 4th and 5th singular vectors are near-degenerate and form an unstable 2D subspace; fix by constraining or eliminating one parameter degree of freedomThe calibration is under-determined; add more option strikes to the objective functionThe numerical solver has not converged; switch from Levenberg-Marquardt to Newton-RaphsonThe singular values σ₄ and σ₅ are small, so they should be set to zero using TSVD with k=3

Quiz: Singular Value Decomposition and Condition Numbers

Quick Quiz