Least Squares Problems Of Several Special Matrix Equations With Rank Constraint And Their Optimal Approximations

Constraint matrix equation has a broad application in vibration theory,network planning,system engineering,civil engineering planning,statistics,economics,imaging and other fields.This paper we studied(anti)Hermite and(anti)Hermite P reflexive Least Square Problem and the best approximation of the matrix equation AX=B with rank constraint. We gained a general least square solution and a expression for the optimal approximation.we studied the following five problems: Problem Ⅰ Given matrices A∈Cm×n,B∈Cm×n,and a non-negative integer s1,Let S1={X|X∈HCn×n,‖AX-B‖=min‖AY-B‖}, Find m1=min rank(X), M1=maxrank(X),及S1s1={X|rank(X)=s1, X∈S1}. Problem Ⅱ Given matrices A∈Cm×n,B∈Cm×n and a non-negative integer s2, Let S2={X|X∈AHCn×n,‖AX-B‖=min‖AY-B‖}, Find m2=min rank(X),M2=maxrank(X),及Ss2={X|rank(X)=s2,X∈S2}. Problem Ⅲ Given matrices A∈Cm×n,B∈Cm×n and a non-negative integer s3, Let S3={X|X∈HCn×n(P),‖AX-B‖=min‖AY-B‖}, Find m3=min rank(X), M3=maxrank(X),及Ss3={X|rank(X)=s3, X∈S3}.Problem Ⅳ Given matrices A∈Cm×n,B∈Cm×n and a non-negative integer s4, Let S4={X|X∈AHCn×n(P),‖AX-B‖=min‖AY-B‖}, Find m4=min rank(X), M4=maxrank(X),及Ss4={X|rank(X)=s4, X∈S4}.Problem Ⅴ Given matrix X∈Cn×n,respectively for X∈Smi(i=1,2,3,4),such that‖X-X*‖=min‖X-X*‖.Here‖·‖is the Frobenius norm of the matrix.For the above problems,by using projection theorem,we converted the least squares problem of contradictory matrix equation under rank restriction to the problem of the compatibility matrix equation solution under the restriction of rank.Through the singular value decomposition bllcked Gaussian transform method,we got the solution of Ⅰ-Ⅳ.And on the basis of the minimum rank solution set Smi(i=1,2,3,4)for Ⅰ-Ⅴ, using the invariant Frobenius norm of unitary matrix to and basic properties of this norm, we got a further expression to the solution of problem V.