The Canonical Forms
The Canonical Forms
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发表时间：
2009
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通讯作者：
D. O’Regan
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作者：
R. Agarwal;D. O’Regan
回想一下，在第7.5节的开头，我们指出T∞L(V)的标准形仅仅是矩阵具有特别简单的形式的表示。例如，如果存在T的特征向量的基，则矩阵表示将是对角的。在这种情况下，评估T的各种性质，如它的秩值、行列式和本征值，就变得相当微不足道了。不幸的是，虽然这通常是矩阵表示最理想的形式，但通常也不可能实现。现在我们希望确定T的某些性质，这些性质将使我们能够尽可能多地了解它的矩阵表示可以采用的可能形式。我们在这一章中将考虑三种主要的规范形式：三角形、有理和Jordan。(这不包括Smith表单，它实际上是一个工具，用于找到有理和Jordan表单。)正如我们以前所做的那样，我们的方法将是以不止一种方式研究这些形式。通过这样做，我们将对它们的含义有更多的了解，并学习在数学的各个分支中非常有用的其他技术。8.1初等标准形这一节简单直接地介绍了两个重要结果：复矩阵的三角型和正规矩阵的对角化。首先，假设
Recall that at the beginning of Section 7.5 we stated that a canonical form for T ∞ L(V) is simply a representation in which the matrix takes on an especially simple form. For example, if there exists a basis of eigenvectors of T, then the matrix representation will be diagonal. In this case, it is then quite trivial to assess the various properties of T such as its rank, determinant and eigenval-ues. Unfortunately, while this is generally the most desirable form for a matrix representation, it is also generally impossible to achieve. We now wish to determine certain properties of T that will allow us to learn as much as we can about the possible forms its matrix representation can take. There are three major canonical forms that we will consider in this chapter : triangular, rational and Jordan. (This does not count the Smith form, which is really a tool, used to find the rational and Jordan forms.) As we have done before, our approach will be to study each of these forms in more than one way. By so doing, we shall gain much insight into their meaning, as well as learning additional techniques that are of great use in various branches of mathematics. 8.1 ELEMENTARY CANONICAL FORMS In order to ease into the subject, this section presents a simple and direct method of treating two important results: the triangular form for complex matrices and the diagonalization of normal matrices. To begin with, suppose