Next, T(1) – R*(1) = 0, which proves assertion 2 of Theorem 3. The space F* of vectors F orthogonal to 1 is invariant under V and , thus R* will have K − 1 eigenvectors of F* (assertion 4 of Theorem 3). If B has rank K (which will be true except in special cases), then TDBDT has rank K − 1 and if M(k) → ∞ for each k, then R* will have K − 1 nonzero eigenvalues which become arbitrarily large. More generally, if B has rank r, then the matrix TDBDT will have rank r − 1, and the r − 1 eigenvalues of R* that depend on B, again will become arbitrarily large as M(k) → ∞. Note that the matrix TET which arises from sampling noise is bounded. (In fact TET is a contraction and has all eigenvalues less than 1.) This completes the proof of Theorem 3.
