Block analogies of comparison matrices
For a complex \(m\times n\) matrix \(A\) denote by \(s(A)\) the \(m\times n\) matrix whose diagonal entries are the singular values of \(A\) in descending order. If \(A=(A_{ik})\) denotes a partition of \(A\) into blocks define the block partitioned matrix \(m(A)\) by \(m(A)_{ik}=s(A_{ik})\). Two conformally partitioned matrices are \textit{equimodular} if \(m(A)=m(B)\). For symmetrically \(r\) by \(r\) partitioned \(A\), define the \(r\times r\) matrix \(H_*(A)\) by \((H_*(A))_{ii}=\)minimal singular value of \(A_{ii}\), \((H_*(A))_{ik}=-(\)maximal singular value of \(A_{ik}\)), and a matrix of size of \(A\), \(H(A)\) by \(H(A)_{ii}=s'(A_{ii})\), \(H(A)_{ik}=-s(A_{ik})\) where \(s'(A_{ii})\) is obtained from \(s(A_{ii})\) writing the diagonal entries in reversed order \((i\neq k)\). Theorem 1: If A is square and - not necessarily symmetrically - block partitioned matrix then are equivalent: (i): Every matrix equimodular with \(A\) is nonsingular. (ii): There exists a permutation of block rows taking \(A\) into a symmetrically partitioned matrix \(B\) such that \(H_*(B)\) is an \(M\)-matrix. Theorem 2: If \(A\) is symmetrically partitioned and \(H(A)\) an \(M\)-matrix, then \(|\det(A)|\geq \det H(A)\). Theorem 1 extends a theorem of \textit{P. Camion} and \textit{A. J. Hoffman} [Pac. J. Math. 17, 211-214 (1966; Zbl 0145.03902)] which is also used in implication \((i)\rightarrow (ii)\) of the proof, while \((ii)\leftarrow (i)\) uses the theory of generalized norms as developed by \textit{M. Fiedler} and \textit{V. Pták} [Czech. Math. J. 12(87), 558-571 (1962; Zbl 0116.25302)]. Theorem 2 generalizes a result of \textit{A. Ostrowski} [Comment. Math. Helv. 10, 69-96 (1937; Zbl 0017.29010)]. Its very interesting proof makes use of a generalization of von Neumann's trace inequality, the relation \(r(B)\leq r(m(B))\) for spectral radii given symmetrically partitioned square \(B\), and power series. The authors use to a good part their methods from Czechosl. Math. 12(87), 382-400 (1962; Zbl 0131.24806) and ibid. 19(94), 428-451 (1969; Zbl 0282.15014)]. Considering a matrix as partitioned into \(1\times 1\) blocks yields the classical results. In this case \(H(A)=H_*(A)\), this being the traditional Hadamard comparison matrix of \(A\).
scientific article; zbMATH DE number 1011338