Using Markov Chains to Reduce Randomness

1 Useful Linear Algebra Let v = (v1, v2, . . . , vn) be a non-zero n-dimensional row vector and P an n× n matrix. • We say v is an eigenvector of P with corresponding eigenvalue λ iff vP = λv. • The L1-norm of v (denoted ‖v‖1) is ∑n i=1 vi. • The L2-norm of v (denoted ‖v‖2) is √∑n i=1 v 2 i . • The inner product of two vectors v and w (denoted v ·w) is ∑n i=1 viwi. • We say vectors v,v, . . .v are orthonormal iff v · v = { 1 if i = j 0 if i 6= j Suppose P is an n × n matrix with positive entries, eigenvectors v,v, . . . ,v, and eigenvalues λ1, λ2, . . . , λn. Let α ∈ R. Using the above definitions we derive the following facts: Fact 1 αP has eigenvectors v,v, . . . ,v and eigenvalues αλ1, αλ2, . . . , αλn Proof v(αP ) = α(vP ) = αλiv . Fact 2 P + I has eigenvectors v,v, . . . ,v and eigenvalues λ1 + 1, λ2 + 1, . . . , λn + 1 Proof v(P + I) = vP + vI = λiv (i) + v = (λi + 1)v .