From fbadeb93c5f2d81f22d9528526988c28caa5e9a8 Mon Sep 17 00:00:00 2001 From: Blinorot Date: Mon, 3 May 2021 13:48:32 +0300 Subject: [PATCH] Lost part of lecture added, fixed reference to part, fixed typos --- Lectures/lecture04.tex | 220 +++++++++++++++++++++++++++++++++++++++++ Lectures/lecture18.tex | 2 +- main.pdf | Bin 2459738 -> 2131978 bytes 3 files changed, 221 insertions(+), 1 deletion(-) diff --git a/Lectures/lecture04.tex b/Lectures/lecture04.tex index f789336..c085de2 100644 --- a/Lectures/lecture04.tex +++ b/Lectures/lecture04.tex @@ -320,6 +320,226 @@ \subsection{Полиномиальное исчисление от матриц} Более того, минимальный многочлен автоматически говорит, когда можно делить на выражение от матрицы, а когда нет. Например, на $A - E$ поделить можно, так как $1$ не является корнем $f$, с другой стороны на матрицы $A \pm\sqrt{3}E$ делить нельзя. +\paragraph{Примеры вычисления минимального многочлена} + +Матрица $A\in\Matrix{n}$ называется верхне треугольной, если $a_{ij} = 0$ при $i > j$, то есть матрица имеет вид +\[ +A= +\begin{pmatrix} + {*}&{*}&{\ldots}&{*}\\ + {0}&{*}&{\ldots}&{*}\\ + {\vdots}&{\vdots}&{\ddots}&{\vdots}\\ + {0}&{0}&{\ldots}&{*} +\end{pmatrix} +\] +То есть элементы под главной диагональю обязательно раны нулю, а остальные какие угодно (может быть тоже нули). Аналогично матрица называется нижне треугольной, если $a_{ij} = 0$ при $i < j$, то есть верхний правый угол матрицы над диагональю заполнен нулями. + +Матрица $A\in \Matrix{n}$ называется нильверхне треугольной, если она верхне треугольная и на диагонали у нее стоят нули, то есть она имеет вид +\[ +\begin{pmatrix} + {0}&{*}&{\ldots}&{*}\\ + {0}&{0}&{\ldots}&{*}\\ + {\vdots}&{\vdots}&{\ddots}&{\vdots}\\ + {0}&{0}&{\ldots}&{0} +\end{pmatrix} +\] + +\begin{claim*} + Пусть $f\in\mathbb R[x]$, тогда + \begin{enumerate} + \item + $ + f + \begin{pmatrix} + {\lambda_1}&{\ldots}&{*}\\ + {\vdots}&{\ddots}&{\vdots}\\ + {0}&{\ldots}&{\lambda_n} + \end{pmatrix} + = + \begin{pmatrix} + {f(\lambda_1)}&{\ldots}&{*}\\ + {\vdots}&{\ddots}&{\vdots}\\ + {0}&{\ldots}&{f(\lambda_n)} + \end{pmatrix} + $ + + \item + $ + \begin{pmatrix} + {0}&{\ldots}&{*}\\ + {\vdots}&{\ddots}&{\vdots}\\ + {0}&{\ldots}&{0} + \end{pmatrix}^n = 0 + $ + \end{enumerate} +\end{claim*} +\begin{proof} + (1). Здесь надо заметить, что + \begin{align*} + \begin{pmatrix} + {\lambda_1}&{\ldots}&{*}\\ + {\vdots}&{\ddots}&{\vdots}\\ + {0}&{\ldots}&{\lambda_n} + \end{pmatrix} + + + \begin{pmatrix} + {\mu_1}&{\ldots}&{*}\\ + {\vdots}&{\ddots}&{\vdots}\\ + {0}&{\ldots}&{\mu_n} + \end{pmatrix} + &= + \begin{pmatrix} + {\lambda_1+\mu_1}&{\ldots}&{*}\\ + {\vdots}&{\ddots}&{\vdots}\\ + {0}&{\ldots}&{\lambda_n+\mu_n} + \end{pmatrix}\\ + \begin{pmatrix} + {\lambda_1}&{\ldots}&{*}\\ + {\vdots}&{\ddots}&{\vdots}\\ + {0}&{\ldots}&{\lambda_n} + \end{pmatrix} + \begin{pmatrix} + {\mu_1}&{\ldots}&{*}\\ + {\vdots}&{\ddots}&{\vdots}\\ + {0}&{\ldots}&{\mu_n} + \end{pmatrix} + &= + \begin{pmatrix} + {\lambda_1\mu_1}&{\ldots}&{*}\\ + {\vdots}&{\ddots}&{\vdots}\\ + {0}&{\ldots}&{\lambda_n\mu_n} + \end{pmatrix} + \end{align*} + Надо понимать, что верхние коэффициенты могли как-то поменяться, но нам все равно как, нас интересует только то, что стоит на диагонали. Так как в записи многочлена $f$ присутствуют операции сложения, умножения и умножения на число, то из этих формул следует первое утверждение. + + (2). Давайте вначале рассмотрим следующее произведение\footnote{Звездочка означает, что данный коэффициент может быть не ноль.} + \[ + \begin{pmatrix} + {0}&{*}&{*}&{*}&{*}\\ + {}&{0}&{*}&{*}&{*}\\ + {}&{}&{0}&{*}&{*}\\ + {}&{}&{}&{0}&{*}\\ + {}&{}&{}&{}&{0}\\ + \end{pmatrix} + \begin{pmatrix} + {0}&{*}&{*}&{*}&{*}\\ + {}&{0}&{*}&{*}&{*}\\ + {}&{}&{0}&{*}&{*}\\ + {}&{}&{}&{0}&{*}\\ + {}&{}&{}&{}&{0}\\ + \end{pmatrix} + = + \begin{pmatrix} + {0}&{0}&{*}&{*}&{*}\\ + {}&{0}&{0}&{*}&{*}\\ + {}&{}&{0}&{0}&{*}\\ + {}&{}&{}&{0}&{0}\\ + {}&{}&{}&{}&{0}\\ + \end{pmatrix} + \] + Методом пристального взгляда устанавливается, что ненулевой треугольный блок съезжает на одну ячейку вправо (надо понимать, что коэффициенты могут измениться как угодно, мы не следим за их точными значениями). Теперь умножим последнюю матрицу на изначальную еще раз + \[ + \begin{pmatrix} + {0}&{0}&{*}&{*}&{*}\\ + {}&{0}&{0}&{*}&{*}\\ + {}&{}&{0}&{0}&{*}\\ + {}&{}&{}&{0}&{0}\\ + {}&{}&{}&{}&{0}\\ + \end{pmatrix} + \begin{pmatrix} + {0}&{*}&{*}&{*}&{*}\\ + {}&{0}&{*}&{*}&{*}\\ + {}&{}&{0}&{*}&{*}\\ + {}&{}&{}&{0}&{*}\\ + {}&{}&{}&{}&{0}\\ + \end{pmatrix} + = + \begin{pmatrix} + {0}&{0}&{0}&{*}&{*}\\ + {}&{0}&{0}&{0}&{*}\\ + {}&{}&{0}&{0}&{0}\\ + {}&{}&{}&{0}&{0}\\ + {}&{}&{}&{}&{0}\\ + \end{pmatrix} + \] + И убедимся, что у результата уже будет $3$ нулевые диагонали. В итоге, за $n-1$ умножение, мы получим, что ненулевой блок уедет за границы матрицы и она останется целиком нулевой. С другой стороны $n-1$ умножение -- это возведение в $n$-ю степень. +\end{proof} + +Как следствие, мы можем предъявить наивный способ искать минимальный многочлен для верхне треугольных матриц. Пусть у нас $A\in\Matrix{n}$ имеет вид +\[ +\begin{pmatrix} + {\lambda_1}&{\ldots}&{*}\\ + {}&{\ddots}&{\vdots}\\ + {}&{}&{\lambda_n}\\ +\end{pmatrix} +\] +Пусть тогда $f(x) = (x-\lambda_1) \ldots (x-\lambda_n)$. Тогда матрица $f(A)$ будет иметь нулевую диагональ по первому пункту утверждения. А значит, по второму пункту, $f(A)^n = 0$. А значит многочлен $f^n$ зануляет $A$.\footnote{На самом деле, можно показать, что $f$ уже зануляет $A$, но пока мы этого сделать не в состоянии. Проверьте это волшебство на матрицах $2$ на $2$.} А так как минимальный многочлен делит $f$, то достаточно искать нужный многочлен среди делителей $f$. То есть минимальный многочлен будет произведением $(x-\lambda_i)$ в каких-то степенях.\footnote{Согласно предыдущему замечанию, достаточно проверить все делители многочлена $(x-\lambda_1)\ldots(x-\lambda_n)$.} + +\paragraph{Блочно верхне треугольные матрицы} +Матрица $A\in\Matrix{n}$ называется блочно верхне треугольной, если она состоит из блоков $A_{ij}\in\MatrixDim{m_{ij}}{n_{ij}}$ и при этом выполнено два условия: (1) диагональные блоки являются квадратными, т.е. $A_{ii}\in\Matrix{n_{ii}}$, (2) блоки ниже главное диагонали равны нулю, т.е. $A_{ij} = 0$ при $i > j$. Для наглядности, следующие матрицы являются блочно верхне треугольными +\[ +\begin{pmatrix} + {*}&{*}&{*}\\ + {0}&{*}&{*}\\ + {0}&{*}&{*}\\ +\end{pmatrix},\quad +\begin{pmatrix} + {*}&{*}&{*}\\ + {*}&{*}&{*}\\ + {0}&{0}&{*}\\ +\end{pmatrix},\quad +\begin{pmatrix} + {*}&{*}&{*}\\ + {0}&{*}&{*}\\ + {0}&{0}&{*}\\ +\end{pmatrix} +\] +а вот эта матрица НЕ является блочно верхне треугольной +\[ +\begin{pmatrix} + {*}&{*}&{*}\\ + {*}&{*}&{*}\\ + {0}&{*}&{*}\\ +\end{pmatrix} +\] + +Аналогично блочной нильверхне треугольной называется матрица $A\in\Matrix{n}$, если она блочно верхне треугольная и ее диагональные блоки равны нулю. Теперь сформулируем блочный аналог предыдущего утверждения, а доказательство оставим в качестве упражнения. + +\begin{claim}\label{claim::PolyOfUpperBlock} + Пусть $f\in\mathbb R[x]$, тогда + \begin{enumerate} + \item Для любой блочной верхне треугольной матрицы выполнено + \[ + f + \begin{pmatrix} + {A_1}&{\ldots}&{*}\\ + {\vdots}&{\ddots}&{\vdots}\\ + {0}&{\ldots}&{A_k}\\ + \end{pmatrix} + = + \begin{pmatrix} + {f(A_1)}&{\ldots}&{*}\\ + {\vdots}&{\ddots}&{\vdots}\\ + {0}&{\ldots}&{f(A_k)}\\ + \end{pmatrix} + \] + \item Для любой блочной нильверхне треугольной матрицы $A\in\Matrix{n}$ с $k$ диагональных блоков $A^k = 0$. + \end{enumerate} +\end{claim} + +Аналогично не блочному случаю это утверждение дает метод поиска минимального многочлена. Пусть у нас матрица $A\in\Matrix{n}$ имеет следующий блочный вид +\[ +\begin{pmatrix} + {A_1}&{\ldots}&{*}\\ + {\vdots}&{\ddots}&{\vdots}\\ + {0}&{\ldots}&{A_k}\\ +\end{pmatrix} +\] +Тогда вначале надо как-то найти зануляющие многочлены для каждого диагонального блока $A_i$. Пусть это будет $f_i$. Положим $f = f_1\ldots f_k$.\footnote{Как следует из первого пункта утверждения выше, достаточно взять наименьшее общее кратное всех $f_i$.} Тогда по первому пункту утверждения $f(A)$ является блочной нильверхне треугольной матрицей, а значит по второму пункту $f(A)^k = 0$. В этом случае минимальный многочлен надо искать среди делителей многочлена $f^k$.\footnote{Как и в неблочном случае, если $f$ был выбран как произведение $f_i$, то тут достаточно рассматривать делители $f$ так как сам $f$ уже должен занулить нашу матрицу.} + + + + \subsection{Матричные нормы} Здесь нас ждет пример первого абстрактного определения. diff --git a/Lectures/lecture18.tex b/Lectures/lecture18.tex index ba2f5b8..4668a8a 100644 --- a/Lectures/lecture18.tex +++ b/Lectures/lecture18.tex @@ -333,7 +333,7 @@ \subsection{Идеальный спектр} Мы знаем, что $0 = f_\text{min}(\varphi) = p(\varphi) h(\varphi)$. Предположим, что $p(\varphi)$ обратим. Тогда в равенстве $p(\varphi)h(\varphi) = 0$ можно сократить на $p(\varphi)$. -Значит $h(\varphi) = 0$, что противоречит минимальности $\varphi$. +Значит $h(\varphi) = 0$, что противоречит минимальности $f_{\text{min}}$. \end{proof} diff --git a/main.pdf b/main.pdf index 1e7024ef797a82c6f115068d19b132d84d461773..9c01b673ffa70c358deaa0a7ee09bcc1c1d4085e 100644 GIT binary patch delta 836085 zcma&MV~{4m79`rXZFAa~wx?~|Hm8lRZQHhO+qP}ny>oYC_r{BTFXH{n%BVke;#B0x zs$7pv1`dz)2Noum7Z#;sqW=v;KD#)y3d2mqNMvVd3B$t!!=UVLZ$iW%XK1PP*9L|` zmWYXw5r#q3!qMq35y$U;P};=S?5{Zy6I%*XImj=-k!DBzCL7$<3yu0aMjbQ?*eG^P z#pEGq{JrW}p*Sk0#5aG)$87cM;`P$j;MimnwF5Z{2P7D&ffydf~Rl^K`zKj z@j9L! z-`{b%lPL@#h&8(4KaWeAK26dks#kSEohQp#@8p?)6D5B4Mz`dvww;>o7M$=G^Cf@> ze+;4O0Z@XICxIA!)kP$3xcga=>Ss+3=GM=}I;P{Ap_gV<2)bwY%9EF7B;Xpr&#M>h z%B@F@=z6Gr(0|V|%e=^KO>(l$+YL{m zv6?7vb7iO6GC%Y3i)P>%9byAqO05vz7$^S0(%!gp$4($GQh2QcEsxZ1z(76Qf;gjb zC9#UEwjFBTS~m3mEX4>aqrbM!&my#yEOH5At7kK@{@%bituI?eUwJxi#th+^cZ?HN 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