线代第六章定义&定理整理（持续更新中）

Chapter 6

6.1 Inner Products and Norms

Definition (inner product).

Let V be a vector space over F. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F, denoted \(⟨x,y⟩\), such that for all x, y, and z in V and all c in F, the following hold:

(a) \(⟨x + z,y⟩ = ⟨x,y⟩ + ⟨z,y⟩.\)

(b) $⟨cx,y⟩=c⟨x,y⟩. $

(c) \(\overline{⟨x, y⟩} = ⟨y, x⟩,\) where the bar denotes complex conjugation.

(d) \(⟨x,x⟩>0\) if \(x \neq 0\).

Definition (conjugate transpose).

Let \(A ∈ M_{m×n}(F)\). We define the conjugate transpose or adjoint of A to be the \(n×m\) matrix \(A^∗\) such that \((A^∗)_{ij} = \overline{A_{ji}}\) for all \(i,j\).

Definition (inner product space).

A vector space \(V\) over \(F\) endowed with a specific inner product is called an inner product space. If \(F = C\), we call V a complex inner product space, whereas if \(F = R\), we call \(V\) a real inner product space.

Definition of some inner products.

Frobenius Inner product: \(\langle A, B\rangle=\operatorname{tr}\left(B^{*} A\right) \text { for } A, B \in M_{n\times n}(F).\)

实际上就是\(\langle A, B\rangle=\sum_{i}\sum_{j}A_{ij}\overline{B_{ij}}\)。

Standard inner product on \(F^n\): \(x=\left(a_{1}, a_{2}, \ldots, a_{n}\right)\) and \(y=\left(b_{1}, b_{2}, \ldots, b_{n}\right)\) in \(\mathrm{F}^{n}\), \(\langle x, y\rangle=\sum_{i=1}^{n} a_{i} \bar{b}_{i}\).

实际上和Frobenius inner product是一个东西。

H of continuous complex-valued functions defined on the interval \([0, 2π]\): \(\langle f, g\rangle=\frac{1}{2 \pi} \int_{0}^{2 \pi} f(t) \overline{g(t)} d t\).

Theorem 6.1.

Let V be an inner product space. Then for x, y, z ∈ V and c ∈ F , the following statements are true.

(a) \(⟨x,y + z⟩\) = \(⟨x,y⟩\) + \(⟨x,z⟩\).

(b) \(⟨x,cy⟩=\overline c⟨x,y⟩\).

(c) \(⟨x,0⟩ = ⟨0,x⟩ = 0\).

(d) \(⟨x,x⟩=0\) if and only if \(x=0\).

(e) If \(⟨x,y⟩=⟨x,z⟩\) for all \(x∈V\), then \(y=z\).

性质(a)和(b)统称conjugate linear，注意不要漏写共轭。

Definition (norm).

Let \(V\) be an inner product space. For \(x ∈ V\), we define the