Lecture 17 : Reduction to Discrete Unitary Matrix ( Step 2 . 1 )

Let μ = {μ1, μ2, . . . , μs} and ν = {ν1, ν2, . . . , νt} be two decreasing sequences of positive rational numbers of lengths s ≥ 1 and t ≥ 1, respectively i.e. μ and ν satisfy μ1 > μ2 > . . . > μs and ν1 > ν2 > . . . > νt. Let m = {m1,m2, . . . ,ms} and n = {n1, n2, . . . , nt} be two sequences of positive integers such that m = ∑s i=1mi and n = ∑t i=1 ni. The rows of B are indexed by x = (x1, x2) where x1 ∈ [s] and x2 ∈ [mx1 ] and the columns of B are indexed by y = (y1, y2) where y1 ∈ [t] and y2 ∈ [ny1 ]. Then, for all x,y, we have