Linear Algebra for Quant Work

Quant finance is applied linear algebra with money attached. A portfolio is a vector; a risk model is a matrix; "what are my biggest risks" is an eigenvalue question. This lesson rebuilds the essentials in those terms.

Vectors and dot products: portfolios and P&L

A portfolio is a weight vector $w = (w_1, \ldots, w_n)$; asset returns over a day form another vector $r$. The portfolio's return is the dot product:

$$R_p = w^\top r = \sum_i w_i r_i$$

Dot products measure alignment: $w^\top r = \|w\|\|r\|\cos\theta$. Two return streams with high positive dot product (after demeaning) are correlated strategies — geometrically, vectors pointing the same way. "Diversification" means choosing vectors that are close to orthogonal.

Matrices: the covariance matrix is the risk model

The object at the center of everything is the covariance matrix $\Sigma$, with $\Sigma_{ij} = \operatorname{Cov}(r_i, r_j)$. Portfolio variance is the quadratic form

$$\operatorname{Var}(R_p) = w^\top \Sigma\, w,$$

which is the two-asset formula ($w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_{12}$) written once for any $n$. Facts interviews expect:

• $\Sigma$ is symmetric and positive semi-definite (PSD): $w^\top \Sigma w \ge 0$ always — variance can't be negative. If your estimated $\Sigma$ isn't PSD, it implies a portfolio with negative variance: a data or estimation bug, and a classic "spot the problem" interview prompt.
• Correlations near ±1 make $\Sigma$ nearly singular — some portfolio has almost zero risk, which optimizers will lever into absurd positions. Near-singularity is why raw sample covariance matrices are never used untreated.

Eigenvalues: what the market's principal risks are

Since $\Sigma$ is symmetric, it diagonalizes: $\Sigma = Q \Lambda Q^\top$ with orthonormal eigenvectors $q_i$ and eigenvalues $\lambda_1 \ge \lambda_2 \ge \cdots \ge 0$. Interpretation:

• Each eigenvector is a portfolio (a linear combination of assets); its eigenvalue is that portfolio's variance.
• The top eigenvector of an equity covariance matrix is, empirically, "everything moves together" — the market factor. The next few look like industries or styles. This is PCA, and it's the mathematical skeleton of factor models: a handful of large eigenvalues carry most of the variance, and the rest is diversifiable noise.
• The trace ($\sum \lambda_i$ = total variance) and the share $\lambda_1 / \sum \lambda_i$ ("how much of the market is one big trade?") are quoted on real desks — that share spikes in crises, when correlations converge and diversification dies.

Solving systems: regression in matrix form

OLS with design matrix $X$ and target $y$ solves the normal equations:

$$\hat\beta = (X^\top X)^{-1} X^\top y$$

Two practical readings. First, $(X^\top X)$ is (up to scaling) the covariance matrix of the predictors — if predictors are collinear it's near-singular, $\hat\beta$ explodes, and you've rediscovered why multicollinearity destabilizes regressions. Second, nobody inverts matrices in practice; libraries solve the system via decompositions (QR, Cholesky) for speed and numerical stability — a fact worth mentioning in any "how would you implement this" follow-up.

The interview versions

• "Why must a covariance matrix be PSD? What would a negative eigenvalue mean?" — A portfolio with negative variance; impossible, so the estimate is broken (usually asynchronous data or more assets than observations).
• "You run PCA on 500 stocks. What does the first component look like, and what fraction of variance might it explain?" — Roughly equal positive weights (the market), often 30–50% in stressed periods.
• "$X^\top X$ is nearly singular — what happens to your regression, and what do you do?" — Coefficients become huge and unstable with opposite signs; fix with regularization (ridge adds $\lambda I$, making the matrix invertible — a one-line answer that bridges directly into the machine-learning course).