Let (X₁, ...,X_k) be a r.vec. with a covariance matrix C. Under which condition on C does there exist a linear combination t₁X₁+...+t_kX_k with zero variance and with not all t’s equal zero?
(Say, you have estimated the covariance matrix of the returns of a collection of securities in a financial market, and you have a good software for matrix operations. What would you do to figure out whether it is possible to arrange a risk-free investment in these random securities?)
Consider, in particular, the case k = 2.