Optimally conditioned block matrices

AbstractThe paper provides a description of optimally conditioned Hermitian positive-definite block matrices, i.e., of matrices A=(Aij)mi,j=1∈Cn×n, n⩾m⩾2, Aii∈Cni×ni,i=1,…,m, such thatk(A)=minD∈Δ(n1,…,nm){k(D*AD)}.Here, Δ(n1,…,nm)⊆Cn×n is the group of nonsingular block diagonal matrices with diagonal blocks of orders ni, i=1,…,m, and k(A) is the spectral condition number of A. The results obtained generalize those for the particular cases m=n and m=2, see [Proc. Amer. Math. Soc. 6 (1955) 340 and Zap. Nauchn. Sem. POMI, 268 (2000) 72], respectively