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zrHl6AmK3;n!*wAkf!RTeDTy6y@`9H%h=();-rHfA;JxcI1tY*CMDRfiim%maGtf;A z#`tY4X7@P?iuT{ZdUoU+6ZPuG6GZP{7tHv~hZ~kHG4xa}a7Dc~2x{ARsdWAS z2;IU%@+orN1T@I2pAQ9E_fq1z30|js>n31+I}PzUH1r{jnHaZI+)DR*`PJMyK1U&# zk?j8t>?#N%75Y=8;Pw#6rqK2fpaAe>@DTp0775+~@Av%2r>8z#g1`1wP=?c!60S46 z{7^-QQ?ENLR_P~i-y7SK8=af+%*g9{oP*gL?J@;^(H+BZ&d`TiP@w0z5TCUQS(5RR z-N(0M{SWO#>lZCRAl8z_Zwxex(tXaV4Ksmpo5`2Hc+p= z11rJdOyp&j{!(Ti4s|5aY0=jZ#j{h5Eq{<7a|Dqe#`a@{kXEEl3Rae}RrdHs&Z}`klAZmQoMvutX;X2ndo6OVd2ggmz`T zs$47`bZ?&TH67YN$h~=Bk$F(+BMX(=-x?LC+DclC+ zEHLofpf$fk59vZr6|3#CWlfLZ#j01^mHmUo8_{^!v&}VDQIZ?}r=^!qMO}#V4%w2T zZ)t#FJlNScJHLDKO&n2Xqa#I(b_U)bpT`I%(h!8zd8}A>+5f;wQu&Bnb{QQDTE>=N zj3thyRSBu{TMg|3?-nwO6Zk$k7pRQQ45kSSMT8U)4W|tWyL1cu!vLqVr1M$05UcL@ z)88s?zYEJ?iQ{8%F31{NhK(l9rG*QrQ(2X4z5A3=9L@LBsX%y4qd)CH5RPR1J<ZHAuUM*Y4}1BFvL*rRz;h?0*fp8+&%tF2X#^_^sRR$87z@}GN1ktsH4$; z@M~mr<(+ayaTwogC*qhywn)AprvmOVjmfm;Hn>1_I;;O^SACO$@Y<<>aZEpYkgzYe z`#cPGlEA5y(%wZt0(CKIQFo-~?2u`A@!#VS z!N*Sf21NG_>gACCu`5+K$V5X(PE7F)J8eA>@g>CIBU9=!uQ12yG`__DpRZOrj)$Ap gTY(EHtL7((?uCdHu>pdY=)8P<=nM=ps 0$. +где $\beta > 0$. \end{enumerate} \end{proposition} \begin{proof} $(1)\Rightarrow (2)$. Пусть $e_1,e_2$~--- ортонормальный базис пространства $V$, и пусть матрица $T$ в этом базисе имеет вид $$ -\begin{pmatrix}a & c\\b & d\end{pmatrix}. +\begin{pmatrix}\alpha & \gamma\\\beta & \delta\end{pmatrix}. $$ -Тогда $||T(e_1)||^2 = a^2 + b^2$, $||T^*(e_1)||^2 = a^2 + c^2$. +Тогда $||T(e_1)||^2 = \alpha^2 + \beta^2$, $||T^*(e_1)||^2 = \alpha^2 + \gamma^2$. По предложению~\ref{prop:normal-operator-equiv} эти числа равны, -откуда $c = \pm b$. Если $c=b$, то $T$ самосопряжен (его матрица -симметричны), поэтому $c = -b$, при этом $b\neq 0$. +откуда $\gamma = \pm \beta$. Если $\gamma=\beta$, то $T$ самосопряжен (его матрица +симметрична), поэтому $\gamma = -\beta$, при этом $\beta\neq 0$. Перемножим теперь матрицы $T$ и $T^*= T^T$ в одном и в другом порядке. Результаты должны -совпасть, но в правом верхнем углу у одной матрицы стоит $bd$, а у -другой $ab$. Значит, $a=d$, и мы получили матрицу нужного вида. +совпасть, но в правом верхнем углу у одной матрицы стоит $\beta\delta$, а у +другой $\alpha\beta$. Значит, $\alpha=\delta$, и мы получили матрицу нужного вида. -$(2)\Rightarrow (3)$. Если в нашем базисе уже $b>0$, то все доказано, +$(2)\Rightarrow (3)$. Если в нашем базисе уже $\beta>0$, то все доказано, а если нет~--- поменяем знак у второго базисного вектора. $(3)\Rightarrow (1)$. Если $T$ имеет указанный вид, то видно, что $T$ @@ -1626,9 +1626,9 @@ $T|_{U^\perp}$ нормален. матрица оператора $T$ блочно-диагональна, причем каждый блок имеет либо размер $1\times 1$, либо размер $2\times 2$ и вид $$ -\begin{pmatrix} a & -b \\ b & a\end{pmatrix}, +\begin{pmatrix} \alpha & -\beta \\ \beta & \alpha\end{pmatrix}, $$ -где $b > 0$. +где $\beta > 0$. \end{enumerate} \end{theorem} \begin{proof} @@ -1717,7 +1717,7 @@ $V$, в котором матрица оператора $a$ диагональ \end{enumerate} \end{theorem} \begin{proof} -Если оператор самосопряженный, кососимметрический, нормальный, то по +Если оператор самосопряженный, кососимметрический, унитарный, то по теореме~\ref{thm:spectral-unitary} существует базис, в котором его матрица диагональна. Если он самосопряжен, то каждый диагональный блок $1\times 1$ самосопряжен, поэтому в нем стоит комплексное число @@ -1740,14 +1740,14 @@ $V$, в котором матрица оператора $a$ диагональ \item Оператор $a$ является кососимметрическим тогда и только тогда, когда существует ортонормированный базис пространства $V$, в котором матрица оператора $a$ имеет блочно-диагональный -вид, и каждый блок выглядит как $(0)$ или $\begin{pmatrix} 0 & -b - \\ b & 0\end{pmatrix}$ для $b\in\mb R$, $\beta > 0$. +вид, и каждый блок выглядит как $(0)$ или $\begin{pmatrix} 0 & -\beta + \\ \beta & 0\end{pmatrix}$ для $\beta\in\mb R$, $\beta > 0$. \item Оператор $a$ является ортогональным тогда и только тогда, когда существует ортонормированный базис пространства $V$, в котором матрица оператора $a$ имеет блочно-диагональный вид, и каждый блок выглядит как $(1)$, $(-1)$ -или $\begin{pmatrix}a&-b\\ b & a\end{pmatrix}$ для -$a,b\in\mb R$, $b > 0$, $a^2 + b^2 = 1$. +или $\begin{pmatrix}\alpha&-\beta\\ \beta & \alpha\end{pmatrix}$ для +$\alpha,\beta\in\mb R$, $\beta > 0$, $\alpha^2 + \beta^2 = 1$. \end{enumerate} \end{theorem} \begin{proof} @@ -1756,18 +1756,18 @@ $a,b\in\mb R$, $b > 0$, $a^2 + b^2 = 1$. матрица блочно-диагональна, с блоками вида $$ \begin{pmatrix} -a & -b\\ -b & a +\alpha & -\beta\\ +\beta & \alpha \end{pmatrix}, $$ где $b>0$. Если он самосопряжен, то каждый диагональный блок самосопряжен, что -для блока $2\times 2$ указанного вида означает, что $b=-b$, +для блока $2\times 2$ указанного вида означает, что $\beta=-\beta$, что невозможно. Поэтому остаются только блоки размера $1\times 1$, что означает диагональность матрицы. Аналогично, из кососимметричности -для блока $2\times 2$ следует, что $a=0$, а для блока $(\lambda)$ +для блока $2\times 2$ следует, что $\alpha=0$, а для блока $(\lambda)$ размера $1\times 1$~--- что $\lambda = 0$. Наконец, из ортогональности -для блока $2\times 2$ следует, что $s^2+b^2=1$, а для блока +для блока $2\times 2$ следует, что $\alpha^2+\beta^2=1$, а для блока $(\lambda)$~--- что $\lambda^2=1$, откуда следует, что $\lambda=\pm 1$. Обратно, если матрица оператора состоит из блоков указанного вида, @@ -1820,7 +1820,7 @@ $||a(v)|| = ||v||$ для всех $v\in V$. \lambda B(u,v) + \ol\lambda\lambda B(v,v). \end{align*} Воспользуемся равенствами $B(a(x),a(x)) = B(x,x)$ и $B(x,y) = - \ol{B(x,y)}$: + \ol{B(y,x)}$: $$ \lambda B(a(u),a(v)) + \ol{\lambda B(a(u),a(v))} = \lambda B(u,v) + \ol{\lambda B(u,v)}. @@ -1838,7 +1838,7 @@ $||a(v)|| = ||v||$ для всех $v\in V$. Пусть $V = \mb R^3$~--- трехмерное вещественное пространство со стандартным эвклидовым скалярным произведением, $a\colon\mb R^3\to\mb R^3$~--- изометрия на $\mb R^3$. Тогда в некотором -ортогональном базисе матрица оператора $a$ имеет вид +ортонормированном базисе матрица оператора $a$ имеет вид $$ \begin{pmatrix} \pm 1 & 0 & 0\\ @@ -1852,7 +1852,7 @@ $$ \end{corollary} \begin{proof} По лемме~\ref{lem:isometry_equiv} оператор $a$ ортогонален. По -теореме~\ref{thm:euclidean_canonical_forms} найдется ортогональный +теореме~\ref{thm:euclidean_canonical_forms} найдется ортонормированный базис $V$, в котором матрица оператора $a$ имеет блочно-диагональный вид, и блоки имеют вид $(\pm 1)$ или $\begin{pmatrix}\cos(\ph)&\sin(\ph)\\-\sin(\ph)&\cos(\ph)\end{pmatrix}$. Если @@ -1866,17 +1866,17 @@ $\begin{pmatrix}\cos(\ph)&\sin(\ph)\\-\sin(\ph)&\cos(\ph)\end{pmatrix}$ \begin{corollary}[Приведение вещественной квадратичной формы к диагональному виду при помощи ортогонального преобразования] Пусть $(V,B)$~--- эвклидово пространство, и пусть -$q\colon V\times V\to B$~--- симметрическая билинейная +$q\colon V\times V\to \mb R$~--- симметрическая билинейная форма. Существует ортогональный базис пространства $V$, в котором матрица Грама формы $q$ имеет диагональный вид. \end{corollary} \begin{proof} Выберем некоторый ортонормированный базис $\mc B$ пространства $V$; пусть $Q$~--- матрица Грама формы $q$ в этом базисе. -Поскольку форма $q$ симметрична, матрица $Q$ является симметричной +Поскольку форма $q$ симметрична, матрица $Q$ является симметрической матрицей: $Q^T = Q$. Рассмотрим $Q$ как матрицу некоторого оператора $a$ на пространстве $V$; по предложению~\ref{prop:adjoint_matrix} -оператор $q$ самосопряжен. +оператор $a$ самосопряжен. По теореме~\ref{thm:euclidean_canonical_forms} существует ортонормированный базис $\mc C$ пространства $V$, в котором матрица оператора $a$ диагональна. Это означает, что @@ -1932,6 +1932,7 @@ $\lambda_1,\dots,\lambda_m$, и кратность $\lambda_i$ равна раз \subsection{Положительно определенные операторы} \literature{[F], гл. XIII, \S~4, п. 4; [K2], гл. 3, \S~3, пп. 8, 9.} +\nopagebreak Пусть $(V,B)$~--- эвклидово или унитарное пространство, $a\colon V\to V$~--- самосопряженный оператор на нем. @@ -1984,11 +1985,11 @@ $\lambda_1,\dots,\lambda_n$ на диагонали. По теоремам~\ref{thm:unitary_canonical_forms} и~\ref{thm:euclidean_canonical_forms} мы уже знаем, что $a$ самосопряжен. Разложим произвольный вектор $v$ по базису $\mc B$: -$v = \sum_i c_i e_i$. -Тогда $a(v) = \sum_i c_i a(e_i) = \sum_i c_i\lambda_i e_i$. +$v = \sum_i e_i c_i$. +Тогда $a(v) = \sum_i a(e_i) c_i = \sum_i e_i c_i\lambda_i$. Поэтому $$ -B(a(v),v) = B(\sum_i c_i\lambda_i e_i,\sum_j c_i e_j) +B(a(v),v) = B(\sum_i e_i c_i\lambda_i,\sum_j e_j c_i) = \sum_{i,j}\overline{c_i}\lambda_i c_j B(e_i,e_j) = \sum_i\lambda_i \overline{c_i}c_i B(e_i,e_i) = \sum_i\lambda_i |c_i|^2 \geq 0. @@ -2120,7 +2121,7 @@ p^{-1}p^2 p^{-1} = \id$, что и требовалось. Наконец, если $pu = a = p'u'$, то $(pu)^* = (p'u')^*$, откуда $u^* p = (u')^*p'$. Из этого следует, что -$(pu)(u^*p) = (p'u')((u')^*p^*)$, откуда $p^2 = (p')^2$, и в силу +$(pu)(u^*p) = (p'u')((u')^*p')$, откуда $p^2 = (p')^2$, и в силу единственности извлечения квадратного корня (теорема~\ref{thm:square_root_positive}), получаем, что $p=p'$, и, стало быть, $u=u'$.