A brief review of Matrix Analysis by R. A. Horn

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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 $f$.

Vector: An n-element array formed by element from corresponding filed $f$. Denoted as $V$.

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 $\{0\}$, $V$.

Proper Subspace: Subspace except $V$.

Generating Subspace

Definition: The intersection of all subspace of $V$ that contain $S$ is the generating subspace of $V$, denoted as $spanS$. The generating subspace of null vector set $span\{\}={0}$.

Proposition: $spanS$ equals to all linear combination of vectors in $S$.

Sum of Space : The sum of space $S_1$ and $S_2$:

$S_1 + S_2 = span{S_1\cup S_2}$

0.1.5-0.1.6 Basis

Definition

A list of linearly independent vectors from vector space $V$, which take $V$ as the generating subspace of them is a basis of of $V$.

Proposition

• A list of vectors which generate $V$ are one basis of $V$, only when any of its proper subset can no generate $V$.
• A list of linearly independent vectors are one basis of $V$ only when any of the proper subset of $V$ that contains the list can not be linearly independent.
• Every element in $V$ can be represented by basis in a sole way.
• Any list of linearly independent vector in $V$ can be expand to a basis of $V$ 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 $V$ is the dimension of $V$, denoting as $dim V$.
For infinite-dimensional space $V$, there exist a one-to-one relationship for elements in any two basis of $V$.
.

Proposition

$$\begin{split} dim \mathbf{R}^n &= n \ \ dim \mathbf{C}^n &=\begin{cases}n&\mathbf{F}=\mathbf{S}\2n&\mathbf{F}=\mathbf{R}\end{cases}\end{split}$$

$R$, $C$ and $S$ refer respectively to real number field, complex number field and pure complex number field.

Subspace intersection lemma

$dim(S_1 \cap S_2) + dim(S_1 + S_2) = dimS_1 + dimS_2$

$Then$

$dim(S_1 \cup S_2) = dimS_1 + simS_2 - dim(S_1 + S_2) \geq dimS_1 + dim S_2 - dimV$

0.1.8 Isomorphism

Definition

$U$, $V$ are vector space in field $F$,
$F: U \to V$ is an invertible function, that for $x, y \in V, a,b \in F:$

$f(ax+by) = af(x) + bf(y)$

Then $f$ is a isomorphism; $U$, $V$ are isomorphic.

Proposition

Two vector space in one field are isomorphic only when they have same dimension

Example of building a $f$

$If$
$V$ is a n-dimension space in $F$, $\beta={x_1,\cdots ,x_n}$, $x=a_1 x_1+ \cdots +a_n x_n$, $a_i \in F$, $x_i \in V$

$Define$

$\lbrack x\rbrack _\beta = \lbrack a_1, \cdots , a_n \rbrack ^T$

$Then$
for arbitrary $\beta$, $f:x \to \lbrack x \rbrack _\beta$ is one isomorphic.

0.2 Matrix

Denote $A= \lbrack a_{ij} \rbrack \in M_{m,n} (F)$ as a $m \times n$ matrix in field $F$; while $a_{ij}$ denote its element.

0.2.2-0.2.3 Linear transformation

Definition

$If$
$U$, $V$ respectively are n-d space and m-d space in $F$, $\beta _U$, $\beta _V$ are basis of $U$, $V$;
$T$ is a linear transformation that maps $x$, $y$ in $U$, $V$ to $\lbrack x \rbrack _{\beta _U}$ as n-d vectors, $\lbrack y \rbrack _{\beta _V}$ as m-d vectors in $F$.

$Then$
construct a $T(x)=y$, which lead to $\lbrack y \rbrack _{\beta V}=A \lbrack x \rbrack _{\beta _U}$.

$A$ is denote as the representation matrix of $T$.
$T$(or corresponding $A$) depend on $\beta _U$, $\beta _V$.

Domain: $F_n$

range: $rangeA=\{y \in F^m : y=\mathbf{A}x \}$, $x \in F^n$

null space & Nullity: $nullspaceA=\{x \in F^n: \mathbf{A}x=0\},\ nullityA = dim\ nullspaceA$

rank: $rankA = dim\ range A$

Rank-Nullity theorem

$dim\ rangeA + dim\ nullspaceA = rankA + nullityA =n$

0.2.5 Conjugate transpose & trace

Definition

$A^{*} = (\bar{A})^T,\ A \in M_{m,n}(\mathbf{C})$ is the conjugate transpose(Hermitian adjoint) of $A$.

If $A^T=-A$, then $A$ is skew symmetric.

If $A^{*} A=I$, then $A$ is unitary.

Division

for any $A \in M_n ( \mathbf{F} )$, ${\exists}\ S(A),\ C(A)$ that
$S(A)=\frac{1}{2} (A + A^T)$ is symmetric.
$C(A)=\frac{1}{2} (A - A^T)$ is skew symmetric.
$A = S(A) + C(A)$

Trace

$tr A = \sum_{k=1}^q a_{kk},\ A=\lbrack a_{ij} \rbrack \in M_{m,n} (\mathbf{F}),\ q=min\{m,n\}$

0.2.7 Column Space & Row Space

$rangeA=spanS$ is denoted as the column space of $A$, while $S$ is the list of column vectors of $A$.

0.4 Rank

0.4.1 Definition

For $A \in M_{m,n}(\mathbf{F})$, $rank A =dim\ range A$ represent the length of the longest linearly independent column or row vector list.

$rank A^T = A$

0.4.2 Linear equations

Solvability

If equation $A x = b$ is solvable, then it is consistent.

$A x = b$ is consistent only when:

$rank \lbrack {A}\quad b \rbrack = rank A$
while $\lbrack A\quad b \rbrack$ refers to augmented matrix.

If $rank\lbrack A\quad b \rbrack = rank A$, then

$dim\ rangeA = dim range \lbrack A\quad b \rbrack\dim\ spanS_{A_{column}} = dim\ spanS _{\lbrack A\quad b \rbrack _{column}}$

then
$b = \sum_{k=1}^n x_k v_k$
$v_k$ denote column vector of $A$ and $$\lbrack x_1, \dots, x_k \rbrack = x$$.

Proposition

If $rank A=k,\ A \in M_{m,n}(\mathbf{F})$, than $k=n-dim\ nullspace A$

0.5 Nonsingularity

0.5.1 Definition

A linear transformation $T$ or corresponding representation matrix $A$ is nonsingular when it only output $0$ when the input equals to 0.

0.5.2 Proposition

• $A \in M_{m,n}(\mathbf{F})$ is singular when $m$ < $n$.
• $A \in M_{m,n}(\mathbf{F})$ is nonsingular when:
• $A^{-1}$ exist or
• $rankA=n$ or
• $detA \ne 0$.

To be continue...

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