@article{Chen2024, 
author = {Yinlan Chen and Yawen Lan},
title = {The best approximation problems between the least-squares solution manifolds of two matrix equations},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {8},
pages = {20939-20955},
keywords = {linear manifold, best approximation, singular value decomposition, canonical correlation decomposition},
url = {https://www.sciopen.com/article/10.3934/math.20241019},
doi = {10.3934/math.20241019},
abstract = {In this paper, we will deal with the following two classes of best approximation problems about the linear manifolds:  Problem 1. Given matrices  A1,B1,C1, and  D1∈Rm×n, find  d(L1,L2)=minX∈L1,Y∈L2‖X−Y‖, and find  X^∈L1,Y^∈L2 such that  ‖X^−Y^‖=d(L1,L2), where  L1={X∈SRn×n| ‖A1X−B1‖=min} and  L2={Y∈SRn×n| ‖C1Y−D1‖=min}.  Problem 2. Given matrices  A2,B2,E2,F2∈Rm×n and  C2,D2,G2,H2∈Rn×p, find  d(L3,L4)=minX∈L3,Y∈L4‖X−Y‖, and find  X~∈L3,Y~∈L4 such that  ‖X~−Y~‖=d(L3,L4), where  L3={X∈Rn×n| ‖A2X−B2‖2+||XC2−D2‖2=min} and  L4={Y∈Rn×n| ‖E2Y−F2‖2+||YG2−H2‖2=min}. We obtain explicit formulas for  d(L1,L2) and  d(L3,L4), and all the matrices in question by using the singular value decompositions and the canonical correlation decompositions of matrices.}
}