Linear Algebra 3: Your Covariance Matrix Is Not Positive Definite
Disclosure: this post was written with the assistance of an AI system (Claude), which wrote the analysis code, ran the experiments and drafted the text. The topic, the constraints, the data choices and the final review are the author’s. Positive semi-definiteness is usually presented as a technical condition on a matrix. It is not: w’Sw is the variance of the portfolio w’x, so a negative eigenvalue is a combination of your variables whose variance the matrix reports as negative, and the eigenvector names it....