Why Linear Algebra Comes Up
Linear algebra is the backbone of quantitative finance: portfolio optimisation, risk attribution, principal component analysis, factor models, and most machine learning techniques rely on matrix decompositions and eigenvalue problems. Quant research interviews at Two Sigma, AQR, BlackRock, DE Shaw, and other systematic firms regularly include linear algebra questions, often as warmups before deeper statistics or ML discussions.
This guide collects 15 worked examples drawn from real interviews. For broader research-track context, see our quant research interview questions and machine learning finance guide.
Section 1: Eigenvalues and PCA (Questions 1-7)
1. Eigenvalue intuition
What does an eigenvalue tell you about a matrix?
Answer: An eigenvalue λ of matrix A satisfies (Av = λv) for some non-zero vector v (the eigenvector). It represents how much the matrix scales vectors in the direction of its eigenvectors. For symmetric matrices, all eigenvalues are real; for positive-definite matrices, all are positive.
2. PCA in finance
You have monthly returns of 100 stocks over 5 years. Explain how you'd use PCA to find common factors.
Answer: Compute the 100x100 covariance matrix of returns. Find its eigendecomposition. The largest eigenvalue (with corresponding eigenvector) is the first principal component - typically interpreted as "the market factor" (all 100 stocks loading positively). The second PC often captures sector divergence; the third regional or style factors. Together, the first 5-10 PCs typically explain 50-70% of total variance.
3. PCA vs factor model
What's the difference between PCA and a fundamental factor model?
Answer: PCA derives factors from the data (the eigenvectors of the covariance matrix). Fundamental factor models specify factors a priori (market, size, value, momentum, etc.). PCA factors are statistically optimal but uninterpretable; fundamental factors are interpretable but may miss latent structure.
4. Covariance matrix estimation
Computing the sample covariance of 500 stocks from 60 months of data. What's the problem, and what's the solution?
Answer: With 60 observations and 500 dimensions, the sample covariance is singular (rank ≤ 60). Solutions: (1) Shrinkage estimators (Ledoit-Wolf) - shrink the sample toward a structured target like the identity. (2) Factor models - estimate covariance via a low-rank factor structure plus diagonal idiosyncratic risk. (3) Random matrix theory - filter out the noise eigenvalues.
5. PCA computation
Given a 100x100 covariance matrix, how do you compute its eigendecomposition? Time complexity?
Answer: Standard methods: QR algorithm (O(n³)), Jacobi rotations (O(n³)), or for sparse matrices, Lanczos / Arnoldi iteration (faster if you only need top-k eigenvalues). For 100x100, all methods are fast (milliseconds). For 10000x10000, only iterative methods are feasible if you need full decomposition.
6. Power method
Describe the power method for computing the largest eigenvalue.
Answer: Start with random vector x_0. Iterate: x_{k+1} = A x_k / ||A x_k||. Converges to the eigenvector of the largest-magnitude eigenvalue. The eigenvalue is then (Ax)_i / x_i. Convergence rate depends on the ratio λ_2/λ_1 - faster when there's a clear gap. Useful for very large matrices when only the top eigenvector matters.
7. Negative eigenvalues
Your sample covariance matrix has a negative eigenvalue. What does this mean, and how do you fix it?
Answer: Sample covariance shouldn't have negative eigenvalues - it indicates the matrix isn't positive-semi-definite. Cause is usually numerical (e.g., near-singular matrix with rounding errors). Fixes: (1) Project to the nearest PSD matrix (set negative eigenvalues to zero, recompose). (2) Add small λI to the diagonal (regularisation). (3) Use a covariance estimator that's PSD by construction (factor model).
Section 2: Matrix Decompositions (Questions 8-12)
8. SVD
What is singular value decomposition?
Answer: Any m×n matrix A factors as A = UΣV^T where U is m×m orthogonal, Σ is m×n diagonal (singular values), V is n×n orthogonal. Generalisation of eigendecomposition to non-square matrices. Used in: PCA (symmetric A = covariance), latent factor analysis, matrix completion, and many ML algorithms.
9. Cholesky decomposition
What is it, and when do you use it?
Answer: Cholesky factorises a symmetric positive-definite matrix as A = LL^T where L is lower-triangular. Used for: (1) Solving linear systems Ax = b when A is symmetric PSD (faster than LU). (2) Generating correlated random samples (multiply standard Gaussian by L). (3) Quadratic optimisation (e.g., portfolio optimisation).
10. QR decomposition
What is it, and when do you use it?
Answer: Any m×n matrix A factors as A = QR where Q is m×m orthogonal and R is m×n upper-triangular. Used for: (1) Solving least-squares problems (more numerically stable than computing X^T X directly). (2) QR algorithm for eigendecomposition. (3) Numerically stable Gram-Schmidt.
11. LU decomposition
What is it, and when do you use it?
Answer: A = LU where L is lower-triangular, U is upper-triangular. Used for solving multiple linear systems Ax = b_i (common in finance: same model, many scenarios). Compute LU once, then solve each system in O(n²) instead of O(n³). With partial pivoting (PA = LU) for numerical stability.
12. Why is Cholesky used over LU for covariance matrices?
Answer: Cholesky is faster (~half the operations of LU) and more numerically stable for symmetric PSD matrices. Cholesky doesn't need pivoting (the matrix structure guarantees stability if PSD). LU requires pivoting for general matrices.
Section 3: Practical Applications (Questions 13-15)
13. Portfolio optimisation
Solve the Markowitz minimum-variance portfolio: minimise w^T Σ w subject to 1^T w = 1.
Answer: Lagrangian gives w = (Σ^{-1} 1) / (1^T Σ^{-1} 1). In practice, never invert Σ directly - it's numerically unstable. Solve Σ y = 1 via Cholesky (since Σ is PSD), then normalise: w = y / sum(y). For risk-minimum, this is the "global minimum variance" portfolio.
14. Linear regression normal equations
Solve OLS regression coefficients β for X β = y in the least-squares sense. Why is solving (X^T X) β = X^T y not optimal?
Answer: Forming X^T X squares the condition number of X. For ill-conditioned X, this can amplify numerical errors significantly. Use QR decomposition of X instead: X = QR, then solve R β = Q^T y by back-substitution. Or use SVD for the most numerically robust solution. The normal equations approach should be avoided when X is wide (many features) or near-singular.
15. Factor analysis
You have 1000 stocks' returns and want a 10-factor model. How do you fit it?
Answer: Several approaches. (1) PCA: take top 10 eigenvectors of covariance. (2) Factor analysis (FA): assumes returns = B*F + ε with idiosyncratic noise. Maximum likelihood estimation via EM algorithm. (3) Constrained PCA: orthogonal rotation of PCA factors for interpretability. (4) Pre-specified fundamental factors (market, size, value, etc.) plus residual factors via PCA. The choice depends on whether interpretability matters - PCA is faster and statistically optimal; fundamental approaches are interpretable.
How to Use This Guide
For research-track candidates (Two Sigma, AQR, BlackRock, DE Shaw), all 15 questions are likely fair game. For trader and developer tracks, focus on Sections 1 and 3 - eigenvalues and practical applications appear regularly.
For broader prep:
- Quant interview questions hub
- Quant research interview questions
- Machine learning finance guide
- Statistics for quantitative trading
For firm-specific interview content where linear algebra is tested:
Practise the questions Linear Algebra for Quant Interviews: 15 Real Questions 2026 actually asks
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