Let X be a mean-zero random vector in R d (so E(X) = 0). Let := E(XX T ) be the covariance matrix of X, and suppose its eigenvalues are 1 2 d . Let...

Let X be a mean-zero random vector in Rd (so E(X) = 0). Let Σ := E(XXT) be the covariance matrix of X, and suppose its eigenvalues are λ1 ≥λ2 ≥···≥λd.  Let σ>0 be a positive number.

(a) What are the eigenvalues of Σ + σ2I? (b) What are the eigenvalues of (Σ + σ2I)-2?

In both cases, give your answers in terms of σ and the eigenvalues of Σ.