A brief review of Matrix Analysis by R. A. Horn
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个人学习笔记,记录了部分概念或定理及简要的证明过程;如愿深入学习可阅读原书(我可以分享电子版)
Chapter 1: Prerequisite
The purpose of this chapter is to introduce some helpful concepts and propositions without demonstration, many of which establish foundation for the rest chapter of this book.
0.1 Vector Space
0.1.1-0.1.2 Vector Space
Definition
Field: A set of scalar, denoted as
. Vector: An n-element array formed by element from corresponding filed
. Denoted as . Vector Space: A set of vectors which are closed under addition and multiplication operation, and in addition contain a zero vector.
0.1.3 Subspace & Span
Definition
Nontrivial Subspace: Subspace except
, . Proper Subspace: Subspace except
. Generating Subspace
Definition: The intersection of all subspace of
that contain is the generating subspace of , denoted as . The generating subspace of null vector set . Proposition:
equals to all linear combination of vectors in . Sum of Space : The sum of space
and : 0.1.5-0.1.6 Basis
Definition
A list of linearly independent vectors from vector space
, which take as the generating subspace of them is a basis of of . Proposition
- A list of vectors which generate
are one basis of , only when any of its proper subset can no generate . - A list of linearly independent vectors are one basis of
only when any of the proper subset of that contains the list can not be linearly independent. - Every element in
can be represented by basis in a sole way. - Any list of linearly independent vector in
can be expand to a basis of in some way. (Expand refer to "add one or more vectors to the list".)
0.1.7 Dimension
Definition
The amount of vectors in every basis of
is the dimension of , denoting as .
For infinite-dimensional space, there exist a one-to-one relationship for elements in any two basis of .
.Proposition
$$ \begin{split} dim \mathbf{R}^n &= n \ \ dim \mathbf{C}^n &=\begin{cases}n&\mathbf{F}=\mathbf{S}\2n&\mathbf{F}=\mathbf{R}\end{cases}\end{split} $$ , and refer respectively to real number field, complex number field and pure complex number field. Subspace intersection lemma
0.1.8 Isomorphism
Definition
, are vector space in field ,
is an invertible function, that for Then
is a isomorphism; , are isomorphic. Proposition
Two vector space in one field are isomorphic only when they have same dimension
Example of building a
is a n-dimension space in , , , ,
for arbitrary, is one isomorphic. 0.2 Matrix
Denote
as a matrix in field ; while denote its element. 0.2.2-0.2.3 Linear transformation
Definition
, respectively are n-d space and m-d space in , , are basis of , ;
is a linear transformation that maps , in , to as n-d vectors, as m-d vectors in .
construct a, which lead to . is denote as the representation matrix of .
(or corresponding ) depend on , . Domain:
range:
, null space & Nullity:
rank:
Rank-Nullity theorem
0.2.5 Conjugate transpose & trace
Definition
is the conjugate transpose(Hermitian adjoint) of . If
, then is skew symmetric. If
, then is unitary. Division
for any
, that
is symmetric.
is skew symmetric.
Trace
0.2.7 Column Space & Row Space
is denoted as the column space of , while is the list of column vectors of . 0.4 Rank
0.4.1 Definition
For
, represent the length of the longest linearly independent column or row vector list. 0.4.2 Linear equations
Solvability
If equation
is solvable, then it is consistent. is consistent only when:
whilerefers to augmented matrix. If
, then then
denote column vector of and \( \lbrack x_1, \dots, x_k \rbrack = x \). Proposition
If
, than 0.5 Nonsingularity
0.5.1 Definition
A linear transformation
or corresponding representation matrix is nonsingular when it only output when the input equals to 0. 0.5.2 Proposition
is singular when < . is nonsingular when: exist or or .
To be continue...
- A list of vectors which generate