It is proved that the sum of squared residuals for each AR order can be obtained by the main diagonal elements of upper triangular matrix so the column by column procedure can be used to develop a recursive algorithm for AR modeling and spectral estimation .

可以证明，由 上 三角 阵的主对角元素便可得到各阶AR模型的残差平方和。因此用逐列处理的方法可以构成AR模型化与谱估计的递推算法。

As a consequence we can decompose a positive definite matrix as product of two triangular matrices and decompose a non-singular matrix as product of a orthogonal matrix with a upper triangular matrix .

作为推论，我们可以利用矩阵的初等变换把一个正定矩阵分解为两个三角矩阵的积，把一个非奇异实矩阵分解为一个正交矩阵与一个 上 三角 矩降的乘积。

Maps on 2 × 2 Upper Triangular Matrix Algebras Preserving Idempotence over Fields

域上2×2 上 三角 矩阵代数保幂等的映射

Some necessary and sufficient for the existence of unique two-side inverses of infinite upper ( lower ) triangular matrix on a division ring R are proved by compactness arguments .

用紧致性论证给出了任一体R上无限 上（下） 三角 阵具有唯一双侧逆方阵的充要条件。

The Construction of Gorenstein Projective Modules over Upper Triangular Matrix Artin Algebras

上 三角 矩阵Artin代数上 Gorenstein投射模的构造

Graded and Ungraded Properties of Polynomial Rings and Special Upper Triangular Matrix Rings

多项式环及特殊 上 三角 矩阵环的分次与非分次性质

Block upper triangular matrix has an important significance in practice .

第 三 章主要 讨论分块 上 三角 矩阵 特征 值 的 扰动，分块上 三角 矩阵是一类 常见 的 矩阵，在实践中有着重要的意义。

Improve algebraic algorithm of the feedforward three-layer neural network such that the output matrix of hidden layer is upper triangular matrix whose main diagonal elements are not zero and those matrix is nonsingular .

改进前馈三层神经网络代数算法。改进算法后隐层的输出矩阵是主对角线上的元素全部非零的 上 三角 矩阵，而这样的矩阵是非奇异的。

The elimination process leading to the eventual formation of an upper triangular matrix is then carried out .

于是进行导致最后形成一个 上 三角形 矩阵的消去手续。

Thus the solution to characteristic value of any n-order matrix is transferred to the solution to characteristic value of upper triangular matrix by the elementary transformation of the matrix and the concrete step of the solution is given in this article .

从而把求任意1个n级矩阵的特征值的问题通过初等变换转化为求 上 三角形 矩阵的特征值的问题，并给出了求解的具体步骤。

A ring R is weakly symmetric if and only if for any n the n × n upper triangular matrix ring ( T_n ( R )) is weakly symmetric ;

环R是弱对称环当且仅当R上的n×n 上 三角 矩阵环Tn（R）是弱对称环；

A substance is through the primary transformation into an upper triangular matrix the transformation matrix is a unit lower triangular matrix .

实质上是将A通过初等行变换变成一个 上 三角 矩阵，其变换矩阵就是一个单位下三角矩阵。

Subrings satisfying ZC_n ( ZI_n ) of upper triangular matrix ring

上 三角 矩阵环满足ZCn（ZIn）的子环

In this paper we investigate the matrix perturbation analysis of eigenvalue perturbation theory . Applying previous results some new perturbation bounds are presented for block upper triangular matrix and reducible matrix .

本文主要研究矩阵扰动分析问题中的特征值扰动理论，通过利用已有的一些结果，来得到矩阵子块的扰动界，进而得到关于分块 上 三角 矩阵和可约矩阵的新的扰动界。

Additive maps preserving the lattices of invariant subspaces on upper triangular matrix algebras

上 三角 矩阵代数上的保不变子空间格映射超不变子空间的紧摄动

This article gives a Formula for finding inverse matrices of upper ( down ) triangular matrix .

本文给出了 上（下） 三角形 矩阵的一个求逆公式。

If policy-makers don 't make sure of some judgments they can give the upper triangular matrix or lower triangular matrix of the fuzzy complementary judgment matrix then transform it to reciprocal judgment matrix and get the gradient eigenvector of fuzzy complementary judgment matrix .

决策者在对某些判断没有把握的情况下，可以给出互补判断 矩阵 的 上 三角或下三角部分，转换成互反型判断矩阵，得出互补型判断矩阵的梯度特征向量法。

The Two-side Inverses of Infinite Upper ( Lower ) Triangular Matrix on a Field

域上无限 上（下） 三角 方阵的双侧逆方阵

Compared with the LUD ( Lower and Upper triangular matrix Decomposition ) algorithm which also has high estimation performance the new algorithm has higher resolution under low Signal-to-Noise Ratio ( SNR ) condition or same resolution under high SNR .

与同样具有高性能的LUD（Lowerand Upper triangular matrixDecomposition）算法相比，在低信噪比时新算法的分辨率更高，高信噪比时，二者性能则相似。

All the diagonal elements of the matrix are 1 ( the elements are compared to themselves ) . Comparisons in only the upper triangular matrix are done ; values in the lower triangle matrix are the reciprocal of upper triangular matrix .

矩阵的对角元素全为1（元素与其自身相比）只需要对 上 三角 矩阵进行对比，下三角矩阵的值是上三角值的倒数。

The inverses of a class of special infinite matric over a division ring is studied . Some sufficient conditions for the existence of left ( or right ) and two-side inverses of infinite upper triangular matrix on a division ring are proved by compactness argument .

研究了除环上一类特殊无限方阵的逆方阵.用紧致性论证给出了除环上无限 上 三角 阵具有左（右）逆和双侧逆方阵的充分条件。

The subrings satisfying ZC_n ( ZI_n ) of the m × m upper triangular matrix ring T_m ( R ) over a reduced ring R are studied .

研究了reduced环R上的m×m 阶 上 三角 矩阵环Tm（R）的满足ZCn（ZIn）的子环。

The key of schur method is to exchange the diagonal blocks in the upper pseudo - triangular matrix .

Schur向量法的关键在于交换伪上 三角形 矩阵对角线 上的对角块的位置。