ВВЕДЕНИЕ В ЛИНЕЙНУЮ АЛГЕБРУ / Lla09

\section*{\it ‹ҐЄжЁп 9.}

\noindent{\bf Љў ¤а вЁз­лҐ д®а¬л, Ёе бўп§м б бЁ¬¬ҐваЁз­л¬Ё
¬ ваЁж ¬Ё. ЏаЁўҐ¤Ґ­ЁҐ Єў ¤а вЁз­ле д®а¬ Є Є ­®­ЁзҐбЄ®¬г ўЁ¤г.}

Џгбвм $A$ -- «Ё­Ґ©­л© ®ЇҐа в®а Ё§ $\RR^n$ ў $\RR^n$, $n\in\NN$.

\noindent ЋЇаҐ¤Ґ«Ґ­ЁҐ. Љў ¤а вЁз­®© д®а¬®© ­  $\RR^n$ ­ §лў ов
®в®Ўа ¦Ґ­ЁҐ $q: \RR^n\to\RR$: $$q(\bar x)=(\bar x,A(\bar x)),$$  
®в®Ўа ¦Ґ­ЁҐ $A$ -- «Ё­Ґ©­л¬ ®в®Ўа ¦Ґ­ЁҐ¬, § ¤ ойЁ¬ нвг д®а¬г.
”®а¬  $q$ ­ §лў Ґвбп ­Ґўл஦¤Ґ­­®©, Ґб«Ё $q(\bar{x})\neq 0$ ¤«п
$\bar{x}\neq \bar{0}$.

Џгбвм $\bar{x}=(x_i)$ -- бв®«ЎҐж Є®®а¤Ё­ в ўҐЄв®а  $\bar x$,
$a_{i,j}$ -- н«Ґ¬Ґ­вл ¬ ваЁжл $A$.\\ ‡ ¬Ґз ­ЁҐ. „«п «оЎле
ўҐЄв®а-бв®«Ўж®ў $\bar{x},\bar{y}\in\RR^n$
$$(\bar{x},A(\bar{y}))=\sum_{i=1}^n x_i\sum_{j=1}^n
a_{i,j}y_j=\sum_{j=1}^n \left(\sum_{i=1}^n x_i
a_{i,j}\right)y_j=(\bar{y},A^{\ast}(\bar{x}))=
(A^{\ast}(\bar{x}),\bar{y}).$$ ‚ з бв­®бвЁ, $q(\bar
x)=(\bar{x},A(\bar{x})) =(\bar{x}, A^{\ast} (\bar{x}))$. ’ ЄЁ¬
®Ўа §®¬, ЇаҐ¤бв ў«Ґ­ЁҐ «оЎ®© Єў ¤а вЁз­®© д®а¬л ў ўЁ¤Ґ
$q(\bar{x})=(\bar{x},A(\bar{x}))$ ­Ґ®¤­®§­ з­®: ¬®¦­® ЇаЁўҐбвЁ
¬­®Ј® ЇаЁ¬Ґа®ў ¬ ваЁж $T$, ¤«п Є®в®але
$q(\bar{x})=(\bar{x},A(\bar{x}))=(\bar{x},T(\bar{x}))$. ‘।Ё ўбҐе
в ЄЁе ЇаҐ¤бв ў«Ґ­Ё© ®¤­® ЁЈа Ґв ®б®Ўго а®«м. ќв® ЇаҐ¤бв ў«Ґ­ЁҐ, ў
Є®в®а®¬ ¬ ваЁж  $T$ пў«пҐвбп бЁ¬¬ҐваЁз­®©.

\noindent ЋЇаҐ¤Ґ«Ґ­ЁҐ. Њ ваЁж  $T$ ­ §лў Ґвбп бЁ¬¬ҐваЁз­®©, Ґб«Ё
$T=T^{\ast}$.

\noindent{’Ґ®аҐ¬ }. ‚бпЄ п Єў ¤а вЁз­ п д®а¬  $q$ ¬®¦Ґв Ўлвм
§ ¤ ­  ў ўЁ¤Ґ $q(\bar{x})=(\bar{x},T(\bar{x}))$ б бЁ¬¬ҐваЁз­®©
¬ ваЁжҐ© $T$; ЇаЁ н⮬ ¬ ваЁж  $T$ Ї® д®а¬Ґ $q$ ®ЇаҐ¤Ґ«пҐвбп
Ґ¤Ё­б⢥­­л¬ ®Ўа §®¬.

$\lhd:$ Џгбвм $q(\bar{x})=(\bar{x},A(\bar{x}))$. ’®Ј¤ , Є Є
®в¬ҐзҐ­® ўлиҐ, $q(\bar{x})=(\bar{x},A^{\ast}(\bar{x}))=
((\bar{x},A(\bar{x}))+(\bar{x},A^{\ast}(\bar{x})))/2$. ‚®§м¬Ґ¬
$T=(A+A^{\ast})/2$. Џ®«гзЁ¬, $q(\bar{x})=(\bar{x},T(\bar{x}))$ Ё
$T$ -- бЁ¬¬ҐваЁз­ п ¬ ваЁж . ЏаҐ¤Ї®«®¦Ё¬
$q(\bar{x})=(\bar{x},U(\bar{x}))$ -- ҐйҐ ®¤­® ЇаҐ¤бв ў«Ґ­ЁҐ д®а¬л
$q$ б бЁ¬¬ҐваЁз­®© ¬ ваЁжҐ© $U$. ’®Ј¤  $(\bar{x},(T-U)\bar{x})=0$
¤«п «оЎ®Ј® $\bar{x}$ Ё, §­ зЁв, $$0=(\bar{x}+\bar{y},
(T-U)(\bar{x}+\bar{y}))= (\bar{y},(T-U)(\bar{x}))+
(\bar{x},(T-U)(\bar{y}))=2(\bar{x},(T-U)(\bar{y}))$$ Ё§-§ 
бЁ¬¬ҐваЁЁ ¬ ваЁжл $T-U$. …б«Ё ў н⮬ а ўҐ­б⢥ ў§пвм
$\bar{x}=(T-U)(\bar{y})$, в® Ї®«гзЁвбп, зв®
$(T-U)(\bar{y})=\bar{0}$ -- ­г«Ґў®© ўҐЄв®а-бв®«ЎҐж. Џ®н⮬г
$T(\bar{y})=U(\bar{y})$ ЇаЁ «оЎ®¬ $\bar{y}$. $\rhd$

‚ ¤ «м­Ґ©иҐ¬ Ўг¤Ґ¬ а бб¬ ваЁў вм в®«мЄ® в ЄЁҐ ЇаҐ¤бв ў«Ґ­Ёп
Єў ¤а вЁз­®© д®а¬л, ў Є®в®але $A$ -- бЁ¬¬ҐваЁз­ п ¬ ваЁж .

\noindent{’Ґ®аҐ¬ }. ‚бпЄ п Єў ¤а вЁз­ п д®а¬  $q$ ў ­ҐЄ®в®а®¬
Ї®¤е®¤п饬 Ў §ЁбҐ ¬®¦Ґв Ўлвм § ¤ ­  ў ўЁ¤Ґ $q(\bar{x})=
(\bar{x},T(\bar{x}))$ б ¤Ё Ј®­ «м­®© ¬ ваЁжҐ© $T$. ’ Є®© ўЁ¤
§ ¤ ­Ёп ­ §лў Ґвбп Є ­®­ЁзҐбЄЁ¬ (Ё«Ё ¤Ё Ј®­ «м­л¬) ўЁ¤®¬ § ¤ ­Ёп
Єў ¤а вЁз­®© д®а¬л.

$\lhd:$ „®Є § вҐ«мбвў® Їа®ўҐ¤Ґ¬ ¤«п б«гз п $n=3$, $(x_1,x_2,x_3)$
-- Є®®а¤Ё­ вл ўҐЄв®а  $\bar{x}$; ®ЎйЁ© б«гз © а бб¬ ваЁў Ґвбп Ї®
 ­ «®ЈЁЁ. ђ §ЎҐаҐ¬ ¤ў  ў®§¬®¦­ле ў аЁ ­в . Љ Є Ё ЇаҐ¦¤Ґ $a_{i,j}$
-- н«Ґ¬Ґ­вл ¬ ваЁжл $A$.

1) $a_{i,i}\neq 0$ е®вп Ўл ¤«п ®¤­®Ј® $i\in\{1,2,3\}$. Џгбвм,
бЄ ¦Ґ¬, $a_{1,1}\neq 0$. €¬ҐҐ¬ $$q(\bar{x})=(\bar{x},A(\bar{x}))=
a_{1,1}x_1^2+2x_1(a_{1,2}x_2+a_{1,3}x_3)+a_{2,2}x_2^2+2x_2
a_{2,3}x_3+a_{3,3}x_3^2.$$ ‚ н⮬ ЇаҐ¤бв ў«Ґ­ЁЁ ўл¤Ґ«Ё¬ Ї®«­л©
Єў ¤а в $$q(x)=a_{1,1}(x_1+\frac{a_{1,2}}{a_{1,1}}x_2+
\frac{a_{1,3}}{a_{1,1}}x_3)^2+ g((x_2,x_3)),$$ ®Ў®§­ зЁў
$g((x_2,x_3))$ --- ®бв ўиЁҐбп Ї®б«Ґ нв®Ј® б« Ј Ґ¬лҐ. Ћ­Ё
ЇаҐ¤бв ў«пов Ё§ бҐЎп Єў ¤а вЁз­го д®а¬г ­  ¤ўг¬Ґа­ле ўҐЄв®а е
$(x_2,x_3)$. ‚ў®¤Ё¬ ­®ўлҐ ЇҐаҐ¬Ґ­­лҐ $(y_1,y_2,y_3)$ Ї® Їа ўЁ«г
$y_1=x_1+a_{1,1}^{-1}(a_{1,2}x_2+a_{1,3}x_3)$, $y_2=x_2$,
$y_3=x_3$. ‚ нвЁе ЇҐаҐ¬Ґ­­ле
$q(\bar{y})=a_{1,1}y_1^2+g((y_2,y_3))$.

2) $a_{i,i}=0$ ¤«п $i\in\{1,2,3\}$; Ґб«Ё $q(\bar{x})=0$ ЇаЁ ўбҐе
$\bar{x}$, в® $q(\bar{x})=0 x_1^2+0 x_2^2+0 x_3^2$. Џгбвм
$q(\bar{x})\neq 0$ ЇаЁ Є Є®¬-«ЁЎ® $\bar{x}$. ЏаҐ¤Ї®«®¦Ё¬,
­ ЇаЁ¬Ґа, $a_{1,2}\neq 0$ Ё $$q(\bar{x})=
2x_1a_{1,2}x_2+2x_1a_{1,3}x_3+2x_2 a_{2,3}x_3.$$ Џгбвм
$x_1=y_1+y_2$, $x_2=y_1-y_2$, $y_3=x_3$. ‚ ­®ўле ЇҐаҐ¬Ґ­­ле
$$q(\bar{y})=2a_{1,2}(y_1^2-y_2^2)+(2(y_1+y_2)a_{1,3}+ 2(y_1-y_2)
a_{2,3})y_3$$ Єў ¤а вЁз­ п д®а¬  $q(\bar{y})$ ᮤҐа¦Ёв б« Ј Ґ¬лҐ б
$y_1^2$, $y_2^2$ б ­Ґ­г«Ґўл¬Ё Є®нддЁжЁҐ­в ¬Ё. Џ®н⮬㠪 ­Ґ© ¬®¦­®
ЇаЁ¬Ґ­Ёвм а бб㦤Ґ­Ёп ў аЁ ­в  1) Ё ЇҐаҐ©вЁ Є а бᬮв७Ёо
Єў ¤а вЁз­®© д®а¬л б $n=2$. $\rhd$

’Ґ®аҐ¬  ({\bf ЎҐ§ ¤®Є § вҐ«мбвў }). „«п ўбпЄ®Ј® «Ё­Ґ©­®Ј®
®ЇҐа в®а  $A$ б бЁ¬¬ҐваЁз­®© ¬ ваЁжҐ© ($A=A^{\ast}$) ў
Їа®бва ­б⢥ $\RR^n$ Ё¬ҐҐвбп ®ав®Ј®­ «м­л© Ў §Ёб Ё§ $n$
б®Ўб⢥­­ле ўҐЄв®а®ў нв®Ј® ®ЇҐа в®а .

{‘«Ґ¤бвўЁҐ}. Џгбвм $q(\bar{x})= (\bar{x},A(\bar{x}))$ б
бЁ¬¬ҐваЁз­®© ¬ ваЁжҐ© $A$, $\{\bar{y}^j\}_{j=1}^n$ --
®ав®Ј®­ «м­л© Ў §Ёб Ё§ $n$ б®Ўб⢥­­ле ўҐЄв®а®ў ®ЇҐа в®а  $A$.
’®Ј¤  ў Ў §ЁбҐ $\{\bar{y}^j\}_{j=1}^n$ Єў ¤а вЁз­ п д®а¬ 
$q(\bar{x})$ Ё¬ҐҐв ¤Ё Ј®­ «м­л© ўЁ¤.\\ $\lhd:$ Џгбвм
$\bar{x}=b_1\bar{y}^1+b_2\bar{y}^2+\dots +b_n\bar{y}^n$. ’®Ј¤ 
$$q(\bar{x})=(\bar{x},A(\bar{x}))=
(b_1\bar{y}^1+b_2\bar{y}^2+\dots +b_n\bar{y}^n,
A(b_1\bar{y}^1+b_2\bar{y}^2+\dots +b_n\bar{y}^n))=$$ $$=
(b_1\bar{y}^1+b_2\bar{y}^2+\dots +b_n\bar{y}^n,
b_1A(\bar{y}^1)+b_2A(\bar{y}^2)+\dots +b_nA(\bar{y}^n))=$$ $$=
(b_1\bar{y}^1+b_2\bar{y}^2+\dots +b_n\bar{y}^n,
b_1\lambda_1\bar{y}^1+b_2\lambda_2\bar{y}^2+\dots
+b_n\lambda_n\bar{y}^n)=$$ $$=
b_1^2\lambda_1(\bar{y}^1,\bar{y}^1)+b_2^2\lambda_2(\bar{y}^2,\bar{y}^2)+\dots
+b_n^2\lambda_n(\bar{y}^n,\bar{y}^n)=b_1^2\lambda_1\cdot
|\bar{y}^1|^2+b_2^2\lambda_2\cdot |\bar{y}^2|^2+\dots
+b_n^2\lambda_n\cdot |\bar{y}^n|^2,$$ Ј¤Ґ $(b_1,\dots,b_n)$ --
Є®®а¤Ё­ вл ўҐЄв®а  $\bar{x}$ ў Ў §ЁбҐ $\{\bar{y}^j\}_{j=1}^n$,
$|\bar{y}^j|^2$ -- Єў ¤а в ¤«Ё­л б®Ўб⢥­­®Ј® ўҐЄв®а  $\bar{y}^j$,
$\lambda_j$ -- б®Ўб⢥­­®Ґ §­ зҐ­ЁҐ, ᮮ⢥вбвўго饥 $\bar{y}^j$:
$A\bar{y}^j=\lambda_j\bar{y}^j$. $\rhd$