diff --git a/.DS_Store b/.DS_Store index 9fd3d26..a050bdf 100644 Binary files a/.DS_Store and b/.DS_Store differ diff --git a/Book.pdf b/Book.pdf index e43dc7e..cd5ca69 100644 Binary files a/Book.pdf and b/Book.pdf differ diff --git a/Book.tex b/Book.tex index b60c833..03a7433 100644 --- a/Book.tex +++ b/Book.tex @@ -9,4 +9,5 @@ \include{Book2/Chapter2/Chapter2section6} \include{Book2/Chapter2/Chapter2section7} \include{Book2/Chapter3/section1} +\include{Book2/Chapter3/section2} \end{document} \ No newline at end of file diff --git a/Book2/.DS_Store b/Book2/.DS_Store index 05ea842..d88f630 100644 Binary files a/Book2/.DS_Store and b/Book2/.DS_Store differ diff --git a/Book2/Chapter2/.DS_Store b/Book2/Chapter2/.DS_Store index f8ea745..9f89fed 100644 Binary files a/Book2/Chapter2/.DS_Store and b/Book2/Chapter2/.DS_Store differ diff --git a/Book2/Chapter3/.DS_Store b/Book2/Chapter3/.DS_Store new file mode 100644 index 0000000..4203f3d Binary files /dev/null and b/Book2/Chapter3/.DS_Store differ diff --git a/Book2/Chapter3/section2.pdf b/Book2/Chapter3/section2.pdf new file mode 100644 index 0000000..cb574e1 Binary files /dev/null and b/Book2/Chapter3/section2.pdf differ diff --git a/Book2/Chapter3/section2.tex b/Book2/Chapter3/section2.tex new file mode 100644 index 0000000..9c33e0b --- /dev/null +++ b/Book2/Chapter3/section2.tex @@ -0,0 +1,45 @@ +\ifx\total\undefined +\documentclass{ctexart} +\usepackage{geometry} +\usepackage{amsmath} +\usepackage{amsfonts} +\begin{document} +\fi + +\paragraph{定义} +设$\mathcal{A} \in \mathcal{L}(V),\mathcal{A}^{*}$是$\mathcal{A}$的伴随算子,如果$\mathcal{A} \circ \mathcal{A}^{*}$,则称$\mathcal{A}$是正规(normal)算子.设$\mathcal{A} \in M_{n}(\mathbb{R})$,如果$AA^{t} = A^{t}A$,则称$A$是正规矩阵. + +\paragraph{注} +由定理2.1和第二章定理2.1可知,$\mathcal{A}$是正规算子当且仅当$\mathcal{A}$在某组单位正交基下的矩阵是正规的. + +\paragraph{引理2.1} +设$\mathcal{A} \in \mathbb{R}^{m \times n}$,如果$tr(AA^{t}) = 0$,则$A = O_{m \times n}$ + +\paragraph{引理2.2} +设$W$是$\mathcal{R}$上$n$维线性空间,$n>0$,$\mathcal{A} \in \mathcal{L}(V)$,则$W$有1维或2维不变子空间. + +\paragraph{引理2.3} +设$A \in M_{n}(\mathbb{R})$是正规的,如果 +%matrix begin +$$ + A = + \left[ + \begin{matrix} + A_{1} & A_{2} \\ + 0 & A_{3} + \end{matrix} + \right] +$$,其中$A_{1} \in M_{d}(\mathbb{R}),A_{2} \in \mathbb{R}^{d \times n-d},A_{3} \in M_{n-d}(\mathbb{R}),0