Linear Algebra Done Right

第四章 Polynomials

  • 4.1 \(\mathbb{F}\) 表示 \(\mathbb{R}\) 或者 \(\mathbb{C}\)
  • 4.2 定义 \(\text{Re }{z},\, \text{im }{z}\)，又记为 \(\Re{z},\, \Im{z}\)
  • 4.3 定义 complex conjugate, absolute value 共轭，模

  • 4.5 复数的性质

    • \(z + \overline{z} = 2 \text{Re } z\).
    • \(z - \overline{z} = 2 \text{Im } z\).
    • \(z \overline{z} = \left\vert z \right\vert ^2\).
    • \(\overline{w+z} = \overline{w} + \overline{z}\), \(\overline{wz} = \overline{w} \overline{z}\).
    • \(\overline{\overline{z}} = z\).
    • \(\left\vert \text{Re } z \right\vert \leq \left\vert z \right\vert\) and \(\left\vert \text{im } z \right\vert \leq \left\vert z \right\vert\).
    • \(\left\vert \overline{z} \right\vert = \left\vert z \right\vert\) .
    • \(\left\vert wz \right\vert = \left\vert w \right\vert \left\vert z \right\vert\).
    • \(\left\vert w + z \right\vert \leq \left\vert w \right\vert + \left\vert z \right\vert\).

  • 4.6 多项式的定义

    Recall that a function \(p : \mathbb{F} → \mathbb{F}\) is called a polynomial with coefficients in \(\mathbb{F}\) if there exist \(a_0, \ldots, a_m \in \mathbb{F}\) such that

    \[p(z) = a_0 + a_1z + a_2z^2 + \cdots + a_mz^m\]
  • 4.7 多项式为zero function就是说所有系数都为0
  • 4.8 多项式除法

若 \(p,s \in \mathcal{P}(\mathbb{F}),\, s \neq 0\)，则存在 \(q,r \in \mathcal{P}(\mathbb{F}\) 使

\[p = sq + r \text{ and } \text{deg} r < \text{deg} s\]

证：设 n = deg p, m = deg s, 若 n < m 则直接令q为0，否则：

Define \(T : \mathcal{P}_{n-m}(\mathbb{F}) \times \mathcal{P}_{m-1}(\mathbb{F}) → \mathcal{P}_n(\mathbb{F})\) by:

\[T(q,r) = sq + r\]

说白了就是 p / s = q 余 r 其中余项r的次数比除数s低。

这显然是个线性空间，由T定义，deq sq >= m, deg r = m-1，所以 \(T(q,r) = 0\) 意味着 \(q = 0,\, r = 0\). 所以 \(\text{null }T\) 维度是0，所以 T 是满射，所以原象存在且唯一

  • 4.9 定义 zeros of a polynomial

    A number \(\lambda \in \mathbb{F}\) is called a zero (or root) of a polynomial \(p \in \mathcal{P}(\mathbb{F})\) if \(p(\lambda) = 0\).

  • 4.10 定义 factor

    A polynomial \(s \in \mathcal{P}(\mathbb{F})\) is called a factor of \(p \in \mathcal{P}(\mathbb{F})\) if there exists a polynomial \(q \in \mathcal{P}(\mathbb{F})\) such that \(p = sq\).

  • 4.11

    简单说就是若z是一个根，则

    \[p(z) = (z - \lambda)q(z)\]

    成立

    证略

  • 4.12 一个m阶多项式最多有m个根

  • 4.13 代数基本定理 Fundamental Theorem of Algebra

    Every nonconstant polynomial with complex coefficients has a zero

    这里居然是用Liouville定理证的，表示看不懂，一个自己找的比较基本的证明见。

  • 4.14 复数域上多项式的唯一分解

    \[p(z) = c(z-\lambda_1)\cdots(z-\lambda_m)\]

  • 4.15 Polynomials with real coefficients have zeros in pairs

    实系数多项式的根的共轭也是根

  • 4.16 实系数二次方程的求根公式

  • 4.17 实数域上多项式的唯一分解

    \[p(x) = c(x-\lambda_1)\cdots(x-\lambda_m)(x^2+b_1x+c_1)\cdots(x^2+b_Mx+c_M)\]

    如果 p(x) 有复数根，那么

    \[\begin{aligned} p(x) &= (x-\lambda)(x-\overline{\lambda})q(x) &= (x^2-2(\operatorname{Re } \lambda)x + \left\vert \lambda \right\vert ^2) q(x) \end{aligned}\]

    如果能证明q(x)也是实系数，就可以递归下去。直到p(x)全是实根，再用4.14即可。

    由4.11 q(z) 存在，但是对于全体实数，\((x-\lambda)(x-\overline{\lambda})\) 为非0实数，所以q(x)为实数，由4.7，q(x)的系数都是实数

第五章 Eigenvalues, Eigenvectors, and Invariant Subspaces

前言写的很好，为了研究 \(\mathcal{L}(V) = \mathcal{L}(V,V)\) 上的函数，如果有一个子空间在映射之后还是在相同的子空间，那么这种空间应该专门起个名字。

  • 5.1 Notation \(\mathbb{F},\, V\)

    • \(\mathbb{F}\) denotes \(\mathbb{R}\) or \(\mathbb{C}\).
    • \(V\) denotes a vector space over \(\mathbb{F}\).

  • 5.2 定义 不变子空间 invariant subspace

    若 \(T \in \mathcal{L}(V)\), \(U\) 是 \(V\) 的子空间，若 \(T(U) \subset U\)，则称U在T变换下是不变子空间。

    另一种记法： \(T|_U \in \mathcal{L}(U)\).

  • 5.5 定义 eigenvalue (又叫characteristic value) 特征值

    Suppose \(T \in \mathcal{L}(V)\). A number \(\lambda \in \mathbb{F}\) is called an eigenvalue of \(T\) if there exists \(v \in V\) such that \(v \neq 0\) and \(Tv = \lambda v\).

  • 5.56 有限维来说，

    • \(\lambda\) 是 \(T\) 的特征值等价于
    • \(T-\lambda I\) 不可逆

  • 5.57 定义 eigenvector 特征向量

  • 5.10 Linearly independent eigenvectors

    若 \(T \in \mathcal{L}(V)\)，它的特征值为 \(\lambda_i\)，对应特征向量为 \(v_i\) 那么 \(v_i\) 线性无关

    证明见书，此处好像没限制维度。

  • 5.13 Number of eigenvalues

    Suppose V is finite-dimensional. Then each operator on V has at most dim V distinct eigenvalues.

    由5.10立得

  • 5.14 定义 \(T \vert _U\) 和 \(T / U\)

    若 \(T \in \mathcal{L}(V)\), \(U\) 是 \(V\) 的不变子空间，则

    • The restriction operator:

      \(T \vert_U \in \mathcal{L}(U)\) is defined by

      \[T \vert_U (u) = Tu\]

      for \(u \in U\).

    • The quotient operator \(T / U \in \mathcal{L}(V/U)\) is defined by

      \[(T/U)(v+U) = Tv+U\]

      for \(v \in V\).

  • 5.16 定义 \(T^m\)

    若 \(T \in \mathcal{L}(V)\), \(m\)是正整数，则定义

    • \(T^m = T \cdots T \text{ m个}T\).
    • \(T^0 = I \in \mathcal{L}(V)\).
    • 若\(T\)可逆，则 \(T^{-m} = \left( T^{-1} \right)^m\)

  • 5.17 定义 \(p(T)\)

    若 \(T \in \mathcal{L}(V),\, p \in \mathcal{P}(\mathbb{F}\)，且

    \[p(z) = a_0 + a_1z + a_2z^2 + \cdots + a_mz^m\]

    则定义

    \[p(T) = a_0I + a_1T + a_2T^2 + \cdots + a_mT^m\]

  • 5.19 定义 多项式之积

    If \(p, q \in \mathcal{P}(\mathbb{F})\), then \(pq \in \mathcal{P}(F)\) is the polynomial defined by

    \[(pq)(z) = p(z)q(z)\]

    注：是卷积

  • 5.20

    • \((pq)(T) = p(T)q(T)\) ;
    • \(p(T)q(T) = q(T)p(T)\) .

  • 5.21 Operators on complex vector spaces have an eigenvalue

    Every operator on a finite-dimensional, nonzero, complex vector space has an eigenvalue.

    证见书

  • 5.22 定义 matrix of an operator \(\mathcal{M}(T)\)
  • 5.24 定义 矩阵的对角 (左上到右下)
  • 5.25 定义 上三角矩阵 (左下为0)

  • 5.26 上三角的等价形式

    Suppose \(T \in \mathcal{L}(V)\) and \(v_1,\ldots,v_n\) is a basis of V. 那么以下三个等价

    • \(\mathcal{M}(T,(v_1,\ldots,v_n))\) 是上三角矩阵
    • \(Tv_j \in \operatorname{span}(v_1,\ldots,v_j) \text{ for each } j = 1,\ldots,n\);
    • \(\operatorname{span}(v_1,\ldots,v_j)\) is invariant under \(T\) for each \(j = 1,\ldots,n\)

    比较显然，不证了。

  • 5.27 Over C, every operator has an upper-triangular matrix

    Suppose \(V\) is a finite-dimensional complex vector space and \(T \in \mathcal{L}(V)\). Then T has an upper-triangular matrix with respect to some basis of \(V\).

    证明也比较长，不抄了，大意是找一个递归证，利用特征值找到不变子空间(所以在\(\mathbb{R}\)上不行，因为特征值可能不存在)令 \(U = \text{range }(T- \lambda I)\)，然后由归纳假设，将 \(U\) 的基扩充成 \(V\) 的基， 对于新扩充的基有 \(T v_k = (T - \lambda I) v_k + \lambda v_k\)，有\(Tv_k \in \operatorname{span}(u_1,\ldots,u_m,v_1,\ldots,v_\), 再由5.26得出。

  • 5.30 上三角矩阵可逆等价于对角线上的元素不为0

    书上的证法比较麻烦，事实上，完全可以构造出一个非0向量使 \(Tv = 0\)

  • 5.32 上三角矩阵的特征值在对角线上。
  • 5.34 定义 ** diagonal matrix** 对角矩阵

  • 5.36 定义 ** eigenspace(特征空间), \(E(\lambda, T)\)

    若 \(T \in \mathcal{L}(V)\) 且 \(\lambda \in \mathbb{F}\). The eigenspace of \(T\) corresponding to \(\lambda\), denoted by \(E(\lambda, T)\), is defined by

    \[E(\lambda, T) = \text{null }(T - \lambda I)\]

    \(\lambda\)对应的特征空间是所有对应\(\lambda\)的特征向量组成的空间。

  • 5.38 不同λ 对应的特征空间的和是直和 (特征空间相互垂直) ，它们的维度和小于总空间的维度

    由5.10已经证过。

  • 5.39 定义 diagonalizable 可对角化

    An operator \(T \in \mathcal{L}(V)\) is called diagonalizable if the operator has a diagonal matrix with respect to some basis of \(V\).

  • 5.41 可对角化的等价条件

    • \(T\) 可对角化
    • \(V\) 有由 \(T\) 的特征向量组成的基

    • 存在一维子空间 \(U_1,\ldots,U_n\) of \(V\)，每个在\(T\)下都是不变子空间，且

      \[V = U_1 \oplus \cdots \oplus U_n\]
    • \(V = E(\lambda_1,T) \oplus \cdots \oplus \text{dim }E(\lambda_m,T)\).
    • \(\text{dim }V = \text{dim }E(\lambda_1,T) + \cdots + \text{dim }E(\lambda_m,T)\).

    证不想看了。。

  • 5.44

    如果 \(T\) 有 \(n\) 个不同的特征值，则 \(T\) 可对角化

第六章 内积空间

  • 6.1 Notation \(\mathbb{F},\, V\)

    • \(\mathbb{F}\) denotes \(\mathbb{R}\) or \(\mathbb{C}\).
    • \(V\) denotes a vector space over \(\mathbb{F}\).

  • 6.2 定义 dot product 点积

    对于 \(\mathbb{R}^n\)，定义点积为element-wise积的和

  • 6.3 定义 inner product 内积

    定义内积是\(V\)上满足如下性质的二元函数

    • positivity
      \(\langle v,v \rangle \geq 0\) for all \(v \in V\);
    • definiteness
      \(\langle v,v \rangle = 0 \iff v=0\);
    • additivityin first slot
      \(\langle u+v,w \rangle = \langle u,w \rangle + \langle v,w \rangle\);
    • homogeneity in first slot
      \(\langle \lambda u,v \rangle = \lambda \langle u,v \rangle\);
    • conjugate symmetry
      \(\langle u,v \rangle = \overline{\langle v,u \rangle}\).

  • 6.4 例子

    • Euclidean inner product on \(\mathbb{F}^n\) is defined by
    \[\langle (w_1,\ldots,w_n),(z_1,\ldots,z_n) \rangle = w_1 \overline{z_1} + \cdots + w_n \overline{z_n}\]
    • [-1,1]上的连接实值函数的内积可以定义为
    \[\langle f,g \rangle = \int_{-1}^{1}{f(x)g(x)\operatorname{dx}}\]
    • \(\mathcal{P}(\mathbb{R})\) 上可定义内积为
    \[\langle p,q \rangle = \int_{0}^{\infty}{p(x)q(x)\operatorname{dx}}\]

  • 6.5 定义 内积空间 (inner product space)

    An inner product space is a vector space \(V\) along with an inner product on \(V\).

  • 6.6 Notation \(V\)

    注意：从现在开始 \(V\) 是表示内积空间。

  • 6.7 内积空间的性质

    • For each fixed \(u \in V,\, \langle v,u \rangle\)作为 \(v\) 的函数是 \(V → \mathbb{F}\) 上的线性函数
    • \(\langle 0,u \rangle = 0\).
    • \(\langle u,0 \rangle = 0\).
    • \(\langle u,v+w \rangle = \langle u,v \rangle + \langle u,w \rangle\).
    • \(\langle u,\lambda v \rangle = \overline{\lambda} \langle u,v \rangle\).

    证明略

  • 6.8 定义 norm, \(\| v \|\)

    \[\| v \| = \sqrt { \langle v,v \rangle }\]

  • 6.11 定义 orthogonal

    u,v are called orthogonal if \(\langle u,v \rangle = 0\).

  • 6.12 Orthogonality and 0

    0 和任意向量都正交 0 是唯一和自己正交的向量

  • 6.13 Pythagorean Theorem

    \[\| u + v \|^2 = \| u \|^2 + \| v \|^2\]

    证：由定义。

  • 6.14 An orthogonal decomposition

    若 \(u,v \in V,\, v \neq 0\)，令 \(c = \frac{\langle u,v \rangle}{\| v \|^2}\) and \(w = u - cv\)，则

    \[\langle w,v \rangle = 0\]

  • 6.15 Cauchy-Schwarz Inequality

    \[\left\vert \langle u,v \rangle \right\vert \leq \| u \| \| v \|.\]

    证略

  • 6.17 例子

    • 若 x_1,\ldots,x_n,y_1,\ldots,y_n \in \mathbb{R} 则
    \[\left\vert x_1y_1 + \cdots + x_ny_n \right\vert \leq ( x_1^2 + \cdots + x_n^2 )( y_1^2 + \cdots + y_n^2 )\]
    • 若 \(f,g\)是[-1,1] 上的实值函数，则

    \(\left\vert \int_{-1}^{1}f(x)g(x)\operatorname{dx} \right\vert^2 \leq \left( \int_{-1}^{1} \left( f(x) \right) ^2 \right) \left( \int_{-1}^{1} \left( g(x) \right) ^2 \right)\).

  • 6.18 三角不等式

    \[\| u+v \| \leq \| u \| + \| v \|\]

  • 6.22 Paralelogram Equality

    \[\| u+v \|^2 + \| u-v \|^2 = 2(\|u\|^2 + \|v\|^2)\]

  • 6.23 定义 orthonormal

    单位正交

  • 6.25

    若 \(e_1,\ldots,e_m\) 是 \(V\) 中的单位正交向量，则

    \[\| a_1e_1 + \cdots + a_me_m \|^2 = \left\vert a_1 \right|vert^2 + \cdots + \left\vert a_n \right\vert^2\]
  • 6.26 单位正交的向量是线性无关的
  • 6.28 n个单位正交的向量是一组基

  • 6.30

    若 \(e_1,\ldots,e_n\) 是一组单位正交基，则

    \[\begin{aligned} v &= \langle v,e_1 \rangle + \cdots + \langle v,e_n \rangle e_v \\ \| v \|^2 &= \left\vert \langle v,e_1 \rangle \right\vert ^2 + \cdots + \left\vert \langle v,e_n \rangle \right\vert |2 \end{aligned}\]

  • 6.31 Gram-Schimdt Procedure 格拉姆-施密特正交化

    \[e_j = \frac{v_j - \langle v_j,e_1 \rangle e_1 - \cdots - \langle v_j,e_{j-1} \rangle e_{j-1}}{ \| {v_j - \langle v_j,e_1 \rangle e_1 - \cdots - \langle v_j,e_{j-1} \rangle e_{j-1}} \| }\]

  • 6.33 Example

    把\(1,x,x^2\)在内积为 \(\langle p,q \rangle = \int_{-1}^{1}{p(x)q(x)\operatorname{dx}}\) 的标准正交基找到了。见书。

  • 6.34 Existence of orthonormal basis

    每个有限维内积空间都有一组单位正交基

  • 6.35 一组单位正交向量可以扩充为一组单位正交基

  • 6.37 Upper-triangular matrix with respect to orthonormal basis

    Suppose \(T \in \mathcal{L}(V)\). If T has an upper-triangular matrix with respect to some basis of V, then T has an upper-triangular matrix with respect to some orthonormal basis of V.

    一个线性变成在一组基下是上三角矩阵，就能在一组标准正交基下是上三角矩阵

    证明见书，现在刚看完，一看就懂。

  • 6.38 Schur’s theorem

    有限维复向量空间上(\(\mathbb{C}^n\))，每个线性变换都有一组标准正交基，使在这组基上对应的矩阵是上三角阵 5.27说的是在\(\mathbb{C}^n\)，每个线性变换都有一组基，使在这组基上对应的矩阵是上三角阵，加之以6.37就可以了。

  • 6.39 定义 linear functional

    和3.92定义完全相同

  • 6.42 Riesz Representation Theorem (里斯表示定理)

    若 \(V\) 是有限维， \(\varphi \in \mathcal{L}(V)\)，那么存在唯一 \(u \in V\) 使

    \[\varphi(v) = \langle v,u \rangle\]

    早就理解出来了，证法也类似，用标准正交基证。

  • 6.44 例 Find \(u \in \mathcal{P}(\mathbb{R})\) such that

    \[\int_{-1}^{1}{p(t) \left( \cos(\pi t) \right) \operatorname{dt}} = \int_{-1}^{1}{p(t)u(t) \operatorname{dt}}\]

    从-1到1的积分是 \(\mathcal{P}_2(\mathbb{R})\) 到 \(\mathbb{R}\) 上的线性映射，题目实际上是，给定\(\varphi(p) ,\, p \in \mathcal{P}_2(\mathbb{R})\)，求\(u\) 使 \(\varphi(p) = \langle p,u \rangle\)

  • 6.45 定义 orthogonal complement

    \[U^\perp = \left\{ v \in V : \langle v,u \rangle = 0 \text{ for every } u \in U \right\}\]

  • 6.46 正交补的性质

    • \(U^\perp\) 是 \(V\) 的子空间
    • \(\left\{ 0 \right\} ^\perp = V\).
    • \(V^\perp = \left\{ 0 \right\}\).
    • \(U \cap U^\perp \subset \left\{ 0 \right\}\).
    • \[U \subset W \subset V \implies W^\perp \subset U^\perp\]

  • 6.47

    若 \(U\) 是 \(V\) 的有限维子空间，则

    \[V = U \oplus U|\perp\]

    注意，U是有限维的，V没有限制

  • 6.50 若 \(V\) 是有限维的，\(U\) 是 \(V\) 的子空间，则

    \[\text{dim }U^\perp = \text{dim }V - \text{dim }U\]

  • 6.51

    \[U = \left( U^\perp \right) ^\perp\]

    证明有点混乱，还没看。

  • 6.53 定义 orthogonal projection, \(P_U\)

    若 \(U\) 是 \(V\) 的有限维子空间，V到U的正交投影(orthogonal projection) \(P_U \in \mathcal{L}(V)\) 是：

    \[\text{ For } v \in V, \text{ write } v = u+w, \text{ where }u \in U \text{ and } w \in U^\perp \text{. Then } P_Uv = u\]
  • 6.54 正交投影的性质，略。

  • 6.56 Minimizing the distance to a subspace

    若 \(U\) 是 \(V\) 的有限维正交子空间，则

    \[\| v -P_Uv \| \leq \| v - u \|\]

第七章 内积空间上的Operator(方阵)

  • 7.1 Notation

    • \(\mathbb{F}\) denotes \(\mathbb{R}\) or \(\mathbb{C}\).
    • \(V \text{ and } W\) denote finite-dimensional inner product spaces over F.

  • 7.2 定义 adjoint, \(T^*\)

    设 \(T \in \mathcal{L}(V,W)\), The adjoint of T is the function \(T^* : W → V\) such that

    \[\langle Tv,w \rangle = \langle v,T^*w \rangle\]

  • 7.5 The adjoint is a linear map

    \(\text{If } \mathcal{L}(V,W), \text{ then } T^* \in \mathcal{L}(W,V)\).

  • 7.6 adjoint的性质

    • \((S+T)^* = S^* + T^*\).
    • \[(\lambda T)^* = \overline{\lambda} T^*\]
    • \(\left( T^* \right) ^* = T\).
    • \(I^* = I\).
    • \[(ST)^* = T^*S^*\]

    7.5和7.6都是用6.3和6.7证的。

  • 7.7 Null space and range of T^*

    若 \(T \in \mathcal{L}(V,W)\) 则

    • \(\text{null }T^* = (\text{range }T)^\perp\);
    • \(\text{range }T^* = (\text{null }T)^\perp\);
    • \(\text{null }T = (\text{range }T^*)^\perp\);
    • \(\text{range }T = (\text{null }T^*)^\perp\).
  • 7.8 定义 conjugate transpose 共轭转置

  • 7.10 The matrix of \(T^*\)

    若 \(T \in \mathcal{L}(V,W),\,e_1,\ldots,e_n 基 V,, f_1,\ldots,f_m 基 W\) 则

    \[\mathcal{M} \left( T^*, (f_1,\ldots,f_m), (e_1,\ldots,e_n) \right)\]

    是

    \[\mathcal{M} \left( T, (e_1,\ldots,e_n), (f_1,\ldots,f_m) \right)\]

    的共轭转置

    证明是由定义推出来的。

  • 7.11 定义 self-adjoint (有人叫它Hermitian)

    An operator \(T \in \mathcal{L}(V)\) is called self-adjoint if \(T = T^*\)

  • 7.13 Eigenvalues of self-adjoint operators are real

    Every eigenvalue of a self-adjoint operator is real.

    \[\lambda \| v \| ^2 = \langle \lambda v,v \rangle = \langle Tv,v \rangle = \langle v,Tv \rangle = \langle v,\lambda v \rangle = \overline{\lambda} \| v \|^2\]

  • 7.14 Over \(\mathbb{C}\), \(Tv\) is orthogonal to \(v\) for all \(v\) only for the \(\boldsymbol{0}\) operator

    若 \(V\) 是复内积空间， \(T \in \mathcal{L}(V)\)，则

    \[\langle Tv,v \rangle = 0 \implies T = 0\]

  • 7.15 Over \(\mathbb{C}\), \(\langle Tv,v \rangle\) is real for all v only for self-adjoint operators

    若 \(V\) 是复内积空间， \(T \in \mathcal{L}(V)\)，则

    \[T \text{ is self-adjoint} \iff \langle Tv,v \rangle \in \mathbb{R}\]

    证：

    \[\langle Tv,v \rangle - \overline{\langle Tv,v \rangle} = \langle Tv,v \rangle - \langle v,Tv \rangle = \langle Tv,v \rangle - \langle T^*v,v \rangle = \langle \left( T-T^* \right) v,v \rangle\]

  • 7.16

    若 \(V\) 是实内积空间， \(T \in \mathcal{L}(V)\)，则

    \[\langle Tv,v \rangle = 0 \implies T = 0\]

  • 7.18 定义 normal 正规矩阵

    \(T \in \mathcal{L}(V)\) is normal if

    \[TT^* = T^*T\]

  • 7.20

    \[T \text{ is normal } \iff \|Tv\| = \|T^*v\|\]

    证：

    \[\begin{aligned} T \text{ is normal} & \iff T^*T-TT^* = 0 \\ & \iff \langle \left( T^*T-TT^* \right) v,v \rangle = 0 \\ & \iff \langle T^*Tv,v \rangle = \langle TT^*v,v \rangle \\ & \iff \text{( by T^*'s definition )} \|Tv\|^2 = \|T^*v\|^2 \end{aligned}\]
  • 7.21 对正规矩阵\(T\)，\(T\)和\(T^*\)有相同的特征向量。

7.B 谱定理

  • 7.24 Complex Spectral Theorem

    若 \(\mathbb{F} = \mathbb{C}\) and \(T \in \mathcal{L}(V)\)，那么以下条件等价

    • T is normal (正规矩阵)
    • T 的特征向量可以构成 \(V\) 的标准正交基
    • 对于 \(V\) 的某个标准正交基， \(T\) 是对角矩阵。

    证明见书

  • 7.26 Invertible quadratic expressions

    若 \(T \in \mathcal{L}(V)\) is self-adjoint, \(b,c \in \mathbb{R}\), \(b^2 < 4c\) 则

    \(T^2 + bT + cI\) 可逆

    证明有空抄。。

  • 7.27 Self-adjoint operators have eigenvalues

    Suppose \(V \neq \{ 0 \}\) and \(T \in \mathcal{L}(V)\) is a self-adjoint operator. Then \(T\) has an eigenvalue.

    证明见书，有时间再抄。

  • 7.28 Self-adjoint operators and invariant subspaces

    若 \(T \in \mathcal{L}(V)\) is self-adjoint and \(U\) is a subspace of \(V\) that is invariant under \(T\). Then

    • \(U^\perp\) in invariant under \(T\);
    • \(T \bar _u \in \mathcal{L}(U)\) is self-adjoint;
    • \(T \bar _u^\perp \in \mathcal{L}(U)\) is self-adjoint;

    证明越来越不直观了，见书。

  • 7.29 Real Spectral Theorem

    若 \(\mathbb{F} = \mathbb{R}\) and \(T \in \mathcal{L}(V)\)，那么以下条件等价

    • T is self-adjoint (埃尔米特矩阵) 这里就是对称了。
    • T 的特征向量可以构成 \(V\) 的标准正交基
    • 对于 \(V\) 的某个标准正交基， \(T\) 是对角矩阵。

    证明见书，有空好好理解一下。

  • 7.31 定义 positive operator

    An operator \(T \in \mathcal{L}(V)\) is called positive if \(T\) is self-adjoint and

    \[\langle Tv,v \rangle \geq 0\]

  • 7.33 Definition square root

    R is a square root of T if \(R^2 = T\)

  • 7.35 positive operators的特征

    以下条件等价

    • T is positive;
    • T is self-adjoint and all the eigenvalues of T are nonnegative;
    • T has a positive square root
    • T has a self-adjoint square root;
    • there exists an operator \(R \in \mathcal{L}(V)\) such that \(T = R^*R\)
  • 7.36 Each positive operator has only one positive square orot.

  • 7.37 定义 isometry(等距同构)

    An operator \(S \in \mathcal{L}(V)\) is called an isometry if for all \(v \in V\)

    \[\| Sv \| = \| v \|\]

    即：S perserve norms.

  • 7.42 isometry(等距同构)的性质

    若 \(S \in \mathcal{L}(V)\)，以下条件等价

    • S is an isometry
    • \(\langle Su,Sv \rangle = \langle u,v \rangle\) for all \(u,v \in V\).
    • \(Se_1,\ldots,Se_n\) is orthonormal if \(e_1,\ldots,e_n \in V\) are orthonormal
    • there exists an orthonormal basis \(e_1,\ldots,e_n\) such that \(Se_1, \ldots, Se_n\) is orthonormal
    • \(S^*S = I\).
    • \(SS^* = I\).
    • \(S^*\) is an isometry
    • \[S^{-1} = S^*\]

  • 7.44 Notation \(\sqrt{T}\)

    If T is a positive operator (半正定), then \(\sqrt{T}\) denotes the unique positive square root of \(T\).

  • 7.45 Polar Depomposition

    Suppose \(T \in \mathcal{L}(V)\). Then there exists an isometry \(S \in \mathcal{L}(V)\) such that

    \[T = S \sqrt{ T^*T }\]

    注意，用矩阵表示时 \(S\) 和 \(T\) 不一定是一组基

    证明以后看。

  • 7.49 定义 singular values

    若 \(T \in \mathcal{L}(V)\). The singular values of T are the eigenvalues of \(\sqrt{T^*T}\), with each eigenvalue \(\lambda\) repeated \(\text{dim }E(\lambda, \sqrt{T^*T})\) times.

  • 7.51 Singular Value Decomposition 若 \(T \in \mathcal{L}(V)\) has singular values \(s_1,\ldots,s_n\). Then there exist orthonormas bases \(e_1,\ldots,e_n\) and \(f_1,\ldots,f_n\) of \(V\) such that for all \(v \in V\)

    \[T v = v_1 \langle v,e_1 \rangle f_1 + \cdots + s_n \langle v,e_n \rangle f_n\]

    证明见书。