On the Low-Rank Approximation Arising in the Generalized Karhunen-Loeve Transform

and Applied Analysis 3 Problem 1 into the fixed rank solution of a matrix equation. Finally, we establish an algorithm for solving Problem 1. Lemma 2. AmatrixX ∈ Rm×n is a solution of Problem 1 if and only if it is a solution of the following matrix equation: XBB T = AB T , rank (X) = d. (12) Proof. It is easy to verify that a matrixX ∈ Rm×n is a solution of Problem 1 if and only if X satisfies the following two equalities simultaneously: 󵄩󵄩󵄩󵄩 A − XB 󵄩󵄩󵄩󵄩F = min X∈R m×p ‖A − XB‖ F , (13) rank (X) = d. (14) Since the normal equation of the least squares problem (13) is XBB T = AB T (15) and noting that the least squares problem (13) and its normal equation (15) have the same solution sets, then (13) and (14) can be equivalently written as XBB T = AB T , rank (X) = d (16) which also imply that Problem 1 is equivalent to (12). Remark 3. From Lemma 2 it follows that Problem 1 is equivalent to (12), hence we can solve Problem 1 by finding a fixed rank solution of the matrix equationXBB = AB. Now we will use generalized singular value decomposition (GSVD) to solve (12). Set C = BB T ∈ R p×p , D = AB T ∈ R m×p . (17) The GSVD of the matrix pair (C,D) is given by (see [24]) C = UΣ 1 W, D = VΣ 2 W, (18) where U ∈ O, V ∈ O, W ∈ Rp×p is a nonsingular matrix, k = rank([C, D]), r = rank(C), t = rank(C) + rank(D) − rank([C, D]), and