Pascal Matrices
Pascal Matrices
复制标题
帕斯卡矩阵
DOI：
10.1080/00029890.2004.11920065
复制
发表时间：
2004
期刊：
影响因子：
--
通讯作者：
Gilbert Strang
中科院分区：
文献类型：
--
作者：
Alan Edelman;Gilbert Strang
每一个n次多项式都有n个根；在[0, 1]上的每一个连续函数都能取到它的最大值；每一个实对称矩阵都有一组完备的正交归一特征向量。“一般性定理”是我们所熟知的数学的一个重要部分。我们几乎无法抗拒进一步推广的冲动！去掉假设条件，使定理更严谨、更困难，包含更多的函数，进入希尔伯特空间……这是我们的天性。数学中的另一个极端或许可以被称为“特殊情形”。一个特定的函数、群或矩阵变得特殊。它和其他的一样遵循一般规则。同时它又有一些小的转折，以一种巧妙的方式将熟悉的对象联系起来。这篇论文是关于一个极其特殊的情形。熟悉的对象是帕斯卡三角形。这个小转折始于将二项式系数构成的三角形放入一个矩阵中。三种不同的矩阵——对称矩阵、下三角矩阵和上三角矩阵——能够以一种方便的方式容纳帕斯卡三角形。截断产生n×n的矩阵\(S_n\)、\(L_n\)和\(U_n\)——当\(n = 4\)时这种模式是可见的：
Every polynomial of degree n has n roots; every continuous function on [0, 1] attains its maximum; every real symmetric matrix has a complete set of orthonormal eigenvectors. “General theorems” are a big part of the mathematics we know. We can hardly resist the urge to generalize further! Remove hypotheses, make the theorem tighter and more difficult, include more functions, move into Hilbert space,. . . It’s in our nature. The other extreme in mathematics might be called the “particular case”. One specific function or group or matrix becomes special. It obeys the general rules, like everyone else. At the same time it has some little twist that connects familiar objects in a neat way. This paper is about an extremely particular case. The familiar object is Pascal’s triangle. The little twist begins by putting that triangle of binomial coefficients into a matrix. Three different matrices—symmetric, lower triangular, and upper triangular—can hold Pascal’s triangle in a convenient way. Truncation produces n by n matrices Sn and Ln and Un—the pattern is visible for n = 4:
影响因子：
2.1
作者：
通讯作者：
--