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 ff.

    Vector: An n-element array formed by element from corresponding filed ff. Denoted as VV.

    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}\{0\}, VV.

    Proper Subspace: Subspace except VV.

    Generating Subspace

    Definition: The intersection of all subspace of VV that contain SS is the generating subspace of VV, denoted as spanSspanS. The generating subspace of null vector set span{}=0span\{\}={0}.

    Proposition: spanSspanS equals to all linear combination of vectors in SS.

    Sum of Space : The sum of space S1S_1 and S2S_2:

    S1+S2=spanS1S2 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 VV, which take VV as the generating subspace of them is a basis of of VV.

    Proposition

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

    Proposition

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

    RR, CC and SS refer respectively to real number field, complex number field and pure complex number field.

    Subspace intersection lemma

    dim(S1S2)+dim(S1+S2)=dimS1+dimS2 dim(S_1 \cap S_2) + dim(S_1 + S_2) = dimS_1 + dimS_2

    ThenThen

    dim(S1S2)=dimS1+simS2dim(S1+S2)dimS1+dimS2dimV 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

    UU, VV are vector space in field FF,
    F:UVF: U \to V is an invertible function, that for x,yV,a,bF:x, y \in V, a,b \in F:

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

    Then ff is a isomorphism; UU, VV are isomorphic.

    Proposition

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

    Example of building a ff

    IfIf
    VV is a n-dimension space in FF, β=x1,,xn\beta={x_1,\cdots ,x_n}, x=a1x1++anxnx=a_1 x_1+ \cdots +a_n x_n, aiFa_i \in F, xiVx_i \in V

    DefineDefine

    [x]β=[a1,,an]T\lbrack x\rbrack _\beta = \lbrack a_1, \cdots , a_n \rbrack ^T

    ThenThen
    for arbitrary β\beta, f:x[x]βf:x \to \lbrack x \rbrack _\beta is one isomorphic.

    0.2 Matrix

    Denote A=[aij]Mm,n(F) A= \lbrack a_{ij} \rbrack \in M_{m,n} (F) as a m×nm \times n matrix in field FF; while aija_{ij} denote its element.

    0.2.2-0.2.3 Linear transformation

    Definition

    IfIf
    UU, VV respectively are n-d space and m-d space in FF, βU\beta _U, βV\beta _V are basis of UU, VV;
    TT is a linear transformation that maps xx, yy in UU, VV to [x]βU\lbrack x \rbrack _{\beta _U} as n-d vectors, [y]βV\lbrack y \rbrack _{\beta _V} as m-d vectors in FF.

    ThenThen
    construct a T(x)=yT(x)=y, which lead to [y]βV=A[x]βU\lbrack y \rbrack _{\beta V}=A \lbrack x \rbrack _{\beta _U}.

    AA is denote as the representation matrix of TT.
    TT(or corresponding AA) depend on βU\beta _U, βV\beta _V.

    Domain: FnF_n

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

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

    rank: rankA=dim rangeArankA = dim\ range A

    Rank-Nullity theorem

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

    0.2.5 Conjugate transpose & trace

    Definition

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

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

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

    Division

    for any AMn(F)A \in M_n ( \mathbf{F} ) ,  S(A), C(A){\exists}\ S(A),\ C(A) that
    S(A)=12(A+AT)S(A)=\frac{1}{2} (A + A^T) is symmetric.
    C(A)=12(AAT)C(A)=\frac{1}{2} (A - A^T) is skew symmetric.
    A=S(A)+C(A)A = S(A) + C(A)

    Trace

    trA=k=1qakk, A=[aij]Mm,n(F), q=min{m,n} 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=spanSrangeA=spanS is denoted as the column space of AA, while SS is the list of column vectors of AA.

    0.4 Rank

    0.4.1 Definition

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

    rankAT=Arank A^T = A

    0.4.2 Linear equations

    Solvability

    If equation Ax=bA x = b is solvable, then it is consistent.

    Ax=bA x = b is consistent only when:

    rank[Ab]=rankA rank \lbrack {A}\quad b \rbrack = rank A
    while [Ab]\lbrack A\quad b \rbrack refers to augmented matrix.

    If rank[Ab]=rankArank\lbrack A\quad b \rbrack = rank A, then

    dim rangeA=dimrange[Ab]dim spanSAcolumn=dim spanS[Ab]column dim\ rangeA = dim range \lbrack A\quad b \rbrack\dim\ spanS_{A_{column}} = dim\ spanS _{\lbrack A\quad b \rbrack _{column}}

    then
    b=k=1nxkvk b = \sum_{k=1}^n x_k v_k
    vkv_k denote column vector of AA and \( \lbrack x_1, \dots, x_k \rbrack = x \).

    Proposition

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

    0.5 Nonsingularity

    0.5.1 Definition

    A linear transformation TT or corresponding representation matrix AA is nonsingular when it only output 00 when the input equals to 0.

    0.5.2 Proposition

    • AMm,n(F)A \in M_{m,n}(\mathbf{F}) is singular when mm < nn.
    • AMm,n(F)A \in M_{m,n}(\mathbf{F}) is nonsingular when:
      • A1A^{-1} exist or
      • rankA=nrankA=n or
      • detA0detA \ne 0.

    To be continue...


 

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