关于两个最小二乘解流形的逼近性

L1={X∈Rn×m|f(X)=‖XA1-B1‖2+‖CT1X-DT1‖2=min},L2={Y∈Rn×m|g(Y)=‖YA2-B2‖2+‖CT2Y-DT2‖2=min},其中A1∈Rm×k1,B1∈Rn×k1,C1∈Rn×l1,D1∈Rm×l1,A2∈Rm×k2,B2∈Rn×k2,C2∈Rn×l2,D2∈Rm×l2均为已知矩阵,本文讨论了L1,L2两个线性流形之间的逼近性,给出了d(L1,L2)=minX∈L1,Y∈L2‖X-Y‖的具体表达式.

On the Approximation Between Linear Manifolds

LetL_1={X∈R~(n×m)|f(X)=‖XA_1-B_1‖~2+‖C~(T)_1X-D~T_1‖~2=min},L_2={Y∈R~(n×m)|g(Y)=‖YA_2-B_2‖~2+‖C~(T)_2Y-D~T_2‖~2=min},where A_1∈R~(m×k_1),B_1∈R~(n×k_1),C_1∈R~(n×l_1),D_1∈R~(m×l_1),A_2∈R~(m×k_2),B_2∈R~(n×k_2),C_2∈R~(n×l_2),D_2∈R~(m×l_2) is given. ‖·‖ stands for Frobenius norm. This paper deals with the approximation of two linear manifolds L_1 and L_2, an explicit representation of (d(L_1,L_2))=minX∈L_1,Y∈L_2‖X-Y‖ is given.