信号处理需要的数学（Matrix Algebra讲义）（更新中……）: chapter3.pdf

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信号处理需要的数学（Matrix Algebra讲义）（更新中……）:

这是一份Matrix Algebra讲义（接近图书，待出版），今天发第一章，属于知识回顾的一章
INTRODUCTION
In this chapter we introduce some of the basic concepts of matrix algebra. You
might have encountered these concepts in an earlier course. It is crucial to be
familiar with the concepts and definitions contained in this chapter in order to
fully understand the material on matrix algebra in further chapters. Hence we
encourage the reader to make sure that he/she is familiar with the basic concepts
contained in this chapter.
Matrix algebra is used in almost every branch of science and technology（矩阵代数的重要性）. It
provides a way to mathematically handle multivariate phenomenon which is
especially useful in linear cases.

先收藏，等更新完再下载。

一天只发一章？等发完再下:22bb

In this chapter we use Gaussian elimination to solve the system of equations
represented by the matrix equation

Ax = b,

where  A is an  mxn  matrix and  b  an  m¨vector. This leads to the  LU-
decomposition of A, which is suitable for computer computations. We are left with
a decomposition of the following form,

      A = LU,

where  L  is a lower triangular mxm matrix and U an upper triangular mxn
matrix. This decomposition is used to solve systems of linear equations and to
find the inverse of a matrix. As we shall notice, if A is a band matrix, then the
matrices L and U are band matrices as well. This is a useful and memory-saving
property.

First we concentrate on square matrices and then we discuss the general case of
mxn matrices. The LU-decomposition of a matrix A is gotten by left multiplying it
by so called Gaussian matrices until we have an upper triangular matrix.

Introduction
We use the notation F introduce some basic concepts involving linear spaces. Linear spaces generalize the properties of the three dimensional space we know to n-dimensional spaces.
Why are these spaces needed anyway? Well, in many cases it has certain
advantages to represent some entity as an n-vector. Every element of this vector refers to one property (quality) of this entity and the value of the element tells us how much of this property our entity has. Mastering multiple qualities all at once is challenging for human minds and matrix algebra provides us with exact methods for this.

4.1 Introduction

In this chapter we show that given any high matrix (m  n ) A, that has rank n,
it can be written as a product of an mxm unitary matrices Q and an mxn upper
triangular matrix R:

  A = QR.

QR decomposition has various applications. It is used, for example, to find the
least squares solution for a system of linear equations Ax=b (multiplying by a
unitary matrix Q does not change the norm) and surprisingly the best known
method for finding the eigenvalues of a general matrix is based on  QR
decomposition.

5.1 Introduction

Once we have a basis for a subspace then all we need to know about a vector in
the subspace are its coordinates to know everything about it. If the basis is
changed then the coordinates change as well. Reasons to choose a particular basis

or to change bases will be presented later in this chapter.

6.1 Introduction

n n
= C if nothing else is mentioned. The reason
In this chapter we assume, that F
for this is that the eigenvalues and eigenvectors of a real matrix can be complex.

Introduction

nxn
A matrix  A
  F  can be transformed to the Jordan canonical form using
similarity transformations. Next we try to find the simplest possible !°almost
diagonal!± form for a general mxn  matrix  A  by using unitary transformation
matrices. We end up with a decomposition that has unbelievable theoretical and
practical value, as we shall see.

7.1 Introduction

Not every matrix has enough linearly independent eigenvectors to be
diagonalizable. However by using similarity transformations every square matrix
can be transformed to the Jordan canonical form, which is almost diagonal.

9.1 Introduction

In this chapter we consider the existence and uniqueness of the solution of a
general system of equations with m equations and n variables. Singular value
decomposition has a significant role in our study

本帖最后由 philoman 于 2009-5-11 15:14 编辑

楼主发完了没？是要在国内出版成英文的吗？

好东西顶一下………………………………………………………………………………

多谢多谢！！！

发完了,国外大牛的lecture，大家可以下载了

谢谢楼主分享信号处理归根到底还是要到数学上来呀！

虽然不是很喜欢英文的文献但是还是很感谢楼主的分享

英语的看着很麻烦不过还是谢谢了

楼主辛苦了！

多谢多谢！！！

好东西要顶，多谢楼主辛劳！

这本书叫啥名啊？
啥出版社啊？

好东西顶一下………………………………………………………………………………