In [l], Ballantine also completely

In [l], Ballantine also completely determined those matrices which are expressible as products of two, three, or four positive matrices. We will show that similar results hold for the products of nonnegative matrices. In particular, it is shown that T is the product of two nonnegative matrices if and only if it is similar to a nonnegative one (Theorem 2.2). The proof for this is slightly more involved than the corresponding one for positive products. The
main step is to show that no nilpotent matrix, except the zero one, can be the product of two nonnegative matrices (Lemma 2.1). In the case of three nonnegative matrices, our attempt for a complete characterization has been less successful. We are able to show that any nilpotent matrix is the product of three nonnegative matrices (Corollary 3.4) and obtain a slightly more
general sufficient condition (Theorem 3.3), which is in terms of Ballantine’s characterization of the products of three positive matrices.

結果 (中文) 1:

在 [l] 百龄坛也完全确定这些矩阵所表述的两、 三个或四个正矩阵的产品。我们将显示为非负矩阵的产品举行了类似的结果。尤其是，它显示 T 是两个非负矩阵的乘积，当且仅当它是类似于一个非负 (定理 2.2)。为此证明是比相应的阳性产物稍显复杂。的主要步骤是显示没有幂零矩阵，除零，可以将两个非负矩阵 (外稃 2.1) 的产品。三个非负矩阵，我们完整的表征的尝试一直不太成功。我们就能够显示任何幂零矩阵的三个非负矩阵 (推论 3.4) 产品并获得稍更多一般的充分条件 (定理 3.3)，是在产品的三个正矩阵百龄坛的表征。

結果 (中文) 3:

在[我]，百龄坛也完全确定的矩阵可以表示为两个，三个产品，或四正矩阵。我们将显示类似的结果保持非负矩阵的产品。特别是，它是示出吨是2个非负矩阵的乘积，如果且仅当它是类似于一个非负的（定理2.2）。对这一点的证明是比相应的一个积极的产品所涉及的。的
步骤主要是表明没有幂零矩阵，除零，可非负矩阵乘积（引理2.1）。在三个非负矩阵的情况下，我们的尝试一个完整的表征已经不太成功。我们能够表明，任何幂零矩阵是三的非负矩阵产品（推论3.4），获得一个更
一般的充分条件（定理3.3），这是在三正矩阵产品百龄坛表征。