A Hadamard-Type Bound on the Coefficients of a Determinant of Polynomials (A. J. Goldstein and R. L. Graham)
A Hadamard-Type Bound on the Coefficients of a Determinant of Polynomials (A. J. Goldstein and R. L. Graham)
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多项式行列式系数的哈达玛型界（A. J. Goldstein 和 R. L. Graham）
作者：
O. Lossers
394问题和解决方案，其中M是r r和非奇异的。在下文中，将不失一般性地假设A已经如上所述被划分，因为如果RAC BG是RAC的非负秩因子分解，则A（RrB）（GCr）是A的非负秩因子分解。如果P是具有r列的真实的矩阵，则定义c（p）={Pxlx>= 0}和（P）{x>= OlPx>= 0}。注意，（#（P）和c（p）是多面体锥。一个锥f称为实锥，如果存在秩为r的非奇异矩阵N，使得C（N）_（？）cT和单纯的，如果存在这样一个N，其中等式成立。Theorf，M.若A是m × n秩为r的非负矩阵，其中r满足0< r< min {m，n}，且A按前所述被分块，则A有非负秩分解当且仅当存在一个单锥，满足N（[M，MQ]）_，cT _c（p）.
394 PROBLEMS AND SOLUTIONS where M is r r and nonsingular. In what follows, it will be assumed without loss of generality that A is already partitioned as above, for if RAC BG is a nonnegative rank factorization of RAC, then A(RrB)(GCr) is a nonnegative rank factorization of A.If P is a real matrix with r columns, then define c (p)={Pxlx>= 0} and (P){x>= OlPx>= 0}. Note that (#(P) and c (p) are polyhedral cones. A cone, f will be said to be solid if there exists a nonsingular matrix N of rank r such that C (N) _.,. cT, and simplicial if there exists such an N for which equality holds. THEORF, M. If A is an m x n nonnegative matrix of rank r where r satisfies 0< r< min {m, n} and A is partitioned as stated previously, then A has a non-negative rank factorization ifand only ifthere exists a simplicial cone, satisfying N ([M, MQ]) _, cT _c (p).