where T, is invertible and T2 is ni

where T, is invertible and T2 is nilpotent. If T, is the product of three positive matrices, then T is the product of three nonnegative matrices. Proof. We may assume that T2 is upper triangular. Let m and n be the sizes of T, and T, respectively. We prove our assertion by induction on nstarting with n = m. If n = m, that is, T = T,, then this is trivially true. Assume that the
assertion holds for n - 1 ( > m). Let T’ be the matrix obtained from T by deleting its last row and column. We have

結果 (中文) 1:

其中 T，是可逆，T2 是幂零。如果 T，是三正矩阵的乘积，T 是三个非负矩阵的乘积。证明。我们可以假定，T2 是上部三角形。让 m 和 n 分别为 T 和 T 的大小。我们证明我们断言诱导对 nstarting 与 n = m。如果 n = m，即 T = T，则这是琐细地真实。假设n-1 的主张 (> m)。让 T' 会得到从 T 删去其最后的行和列的矩阵。我们有

結果 (中文) 3:

其中T，是可逆的，T2是幂零的。如果是三个正矩阵的乘积，则是三个非负矩阵的乘积。证明。我们可以假设T2上三角。让我和氮分别是吨，吨，分别。我们证明了我们的论断，诱导nstarting n = m n = m，即T = T，那么这是很真实的。假设的断言持有1（>）。不要从删除其最后一行和列中获得的矩阵。我们有