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⟩. $

(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) $⟨x,cy⟩=\overline c⟨x,y⟩$.

(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

巴特西

线代第六章定义&定理整理（持续更新中）