% (find-LATEX "2023-2-C3-matrizes-definidas.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2023-2-C3-matrizes-definidas.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2023-2-C3-matrizes-definidas.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2023-2-C3-matrizes-definidas.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2023-2-C3-matrizes-definidas.pdf"))
% (defun e () (interactive) (find-LATEX "2023-2-C3-matrizes-definidas.tex"))
% (defun o () (interactive) (find-LATEX "2023-2-C3-matrizes-definidas.tex"))
% (defun u () (interactive) (find-latex-upload-links "2023-2-C3-matrizes-definidas"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2023-2-C3-matrizes-definidas.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
%          (code-eec-LATEX "2023-2-C3-matrizes-definidas")
% (find-pdf-page   "~/LATEX/2023-2-C3-matrizes-definidas.pdf")
% (find-sh0 "cp -v  ~/LATEX/2023-2-C3-matrizes-definidas.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2023-2-C3-matrizes-definidas.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2023-2-C3-matrizes-definidas.pdf")
%   file:///home/edrx/LATEX/2023-2-C3-matrizes-definidas.pdf
%               file:///tmp/2023-2-C3-matrizes-definidas.pdf
%           file:///tmp/pen/2023-2-C3-matrizes-definidas.pdf
%  http://anggtwu.net/LATEX/2023-2-C3-matrizes-definidas.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Piecewise2 ExprDxDy1")
% (find-Deps1-cps   "Caepro5 Piecewise2 ExprDxDy1")
% (find-Deps1-anggs "Caepro5 Piecewise2 ExprDxDy1")
% (find-MM-aula-links "2023-2-C3-matrizes-definidas" "C3" "c3m232md" "c3md")

% «.defs»			(to "defs")
% «.defs-T-and-B»		(to "defs-T-and-B")
% «.defs-caepro»		(to "defs-caepro")
% «.defs-pict2e»		(to "defs-pict2e")
% «.title»			(to "title")
% «.links»			(to "links")
% «.alguns-exemplos-defs»	(to "alguns-exemplos-defs")
% «.alguns-exemplos»		(to "alguns-exemplos")
% «.versao-mega-rapida»		(to "versao-mega-rapida")
% «.eigshows»			(to "eigshows")
%
% «.djvuize»		(to "djvuize")



\documentclass[oneside,12pt]{article}
\usepackage[colorlinks,citecolor=DarkRed,urlcolor=DarkRed]{hyperref} % (find-es "tex" "hyperref")
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pict2e}
\usepackage[x11names,svgnames]{xcolor} % (find-es "tex" "xcolor")
\usepackage{colorweb}                  % (find-es "tex" "colorweb")
%\usepackage{tikz}
%
% (find-dn6 "preamble6.lua" "preamble0")
%\usepackage{proof}   % For derivation trees ("%:" lines)
%\input diagxy        % For 2D diagrams ("%D" lines)
%\xyoption{curve}     % For the ".curve=" feature in 2D diagrams
%
\usepackage{edrx21}               % (find-LATEX "edrx21.sty")
\input edrxaccents.tex            % (find-LATEX "edrxaccents.tex")
\input edrx21chars.tex            % (find-LATEX "edrx21chars.tex")
\input edrxheadfoot.tex           % (find-LATEX "edrxheadfoot.tex")
\input edrxgac2.tex               % (find-LATEX "edrxgac2.tex")
%\usepackage{emaxima}              % (find-LATEX "emaxima.sty")
%
% (find-es "tex" "geometry")
\usepackage[a6paper, landscape,
            top=1.5cm, bottom=.25cm, left=1cm, right=1cm, includefoot
           ]{geometry}
%
\begin{document}

% «defs»  (to ".defs")
% (find-LATEX "edrx21defs.tex" "colors")
% (find-LATEX "edrx21.sty")

\def\drafturl{http://anggtwu.net/LATEX/2023-2-C3.pdf}
\def\drafturl{http://anggtwu.net/2023.2-C3.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}

% (find-LATEX "2023-1-C2-carro.tex" "defs-caepro")
% (find-LATEX "2023-1-C2-carro.tex" "defs-pict2e")

\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"}  % (find-LATEX "dednat6load.lua")

% «defs-T-and-B»  (to ".defs-T-and-B")
\long\def\ColorDarkOrange#1{{\color{orange!90!black}#1}}
\def\T(Total: #1 pts){{\bf(Total: #1)}}
\def\T(Total: #1 pts){{\bf(Total: #1 pts)}}
\def\T(Total: #1 pts){\ColorRed{\bf(Total: #1 pts)}}
\def\B       (#1 pts){\ColorDarkOrange{\bf(#1 pts)}}

% «defs-caepro»  (to ".defs-caepro")
%L dofile "Caepro5.lua"              -- (find-angg "LUA/Caepro5.lua" "LaTeX")
\def\Caurl   #1{\expr{Caurl("#1")}}
\def\Cahref#1#2{\href{\Caurl{#1}}{#2}}
\def\Ca      #1{\Cahref{#1}{#1}}

% «defs-pict2e»  (to ".defs-pict2e")
%L dofile "Piecewise2.lua"           -- (find-LATEX "Piecewise2.lua")
%L --dofile "Escadas1.lua"           -- (find-LATEX "Escadas1.lua")
%L dofile "ExprDxDy1.lua"               -- (find-LATEX "ExprDxDy1.lua")
%L V = MiniV
%L v = V.fromab
\def\pictgridstyle{\color{GrayPale}\linethickness{0.3pt}}
\def\pictaxesstyle{\linethickness{0.5pt}}
\def\pictnaxesstyle{\color{GrayPale}\linethickness{0.5pt}}
\celllower=2.5pt

\pu



%  _____ _ _   _                               
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___ 
%   | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
%   | | | | |_| |  __/ | |_) | (_| | (_| |  __/
%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/      
%
% «title»  (to ".title")
% (c3m232mdp 1 "title")
% (c3m232mda   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Cálculo 3 - 2023.2}

\bsk

Aula 26: matrizes definidas

\bsk

Eduardo Ochs - RCN/PURO/UFF

\url{http://anggtwu.net/2023.2-C3.html}

\end{center}

\newpage

% «links»  (to ".links")
% (c3m232mdp 2 "links")
% (c3m232mda   "links")

{\bf Links}

\scalebox{0.6}{\def\colwidth{14cm}\firstcol{

% (find-books "__alg/__alg.el" "strang" "159" "3 Orthogonality")
% (find-books "__alg/__alg.el" "strang" "196" "Orthogonal Matrices")
% (find-books "__alg/__alg.el" "strang" "260" "5 Eigenvalues and Eigenvectors")
% (find-books "__alg/__alg.el" "strang" "267" "Eigshow")
% (find-books "__alg/__alg.el" "strang" "312" "Every symmetric matrix")
% (find-books "__alg/__alg.el" "strang" "345" "6 Positive Definite Matrices")
% (find-books "__alg/__alg.el" "strang" "355" "Cholesky decomposition")
\par \Ca{Strang4cap3p42} (p.159) 3 Orthogonality
\par \Ca{Strang4cap3p42} (p.196) Orthogonal Matrices
\par \Ca{Strang4cap5p5} (p.260) 5 Eigenvalues and Eigenvectors
\par \Ca{Strang4cap5p12} (p.267) Eigshow
\par \Ca{Strang4cap5p42} (p.312) Every symmetric matrix ... has real eigenvalues
\par \Ca{Strang4cap5p42} (p.312) Its eigenvectors can be chosen to be orthonormal
\par \Ca{Strang4cap6p5} (p.345) 6 Positive Definite Matrices
\par \Ca{Strang4cap6p15} (p.355) Cholesky decomposition

\ssk

% (find-books "__analysis/__analysis.el" "bortolossi" "379" "O polinômio de Taylor de ordem 2")
% (find-books "__analysis/__analysis.el" "bortolossi" "380" "matriz hessiana")
% (find-books "__analysis/__analysis.el" "bortolossi" "383" "11.3. Formas quadráticas e")
\par \Ca{Bort11p15} (p.379) O polinômio de Taylor de ordem 2
\par \Ca{Bort11p16} (p.380) matriz Hessiana
\par \Ca{Bort11p19} (p.383) 11.3 Formas quadráticas e matrizes definidas

\ssk

% (find-books "__analysis/__analysis.el" "stewart-pt" "850" "14.7 Valores Máximo e Mínimo")
\par \Ca{StewPtCap14p64} (p.850) 14.7 Valores Máximo e Mínimo
\par \Ca{StewPtCap14p65} (p.851) Teste da segunda derivada; D(a,b)

}\anothercol{
}}

\newpage


% «alguns-exemplos-defs»  (to ".alguns-exemplos-defs")
% (c3m232mdp 3 "alguns-exemplos-defs")
% (c3m232mda   "alguns-exemplos-defs")
% (c3m232fhp 9 "alguns-exemplos-defs")
% (c3m232fha   "alguns-exemplos-defs")
%L
%L -- (find-angg "LUA/ExprDxDy1.lua" "ExprDxDy-tests-abc")
%L V3.threeD = "2D"
%L x0,y0 = 3,2
%L defabc = function (s) ExprDxDy.from(s):abc():output() end
%L defabc     "Dx^2"
%L defabc   "1+Dx^2"
%L defabc   "2+Dx^2"
%L defabc          "Dy^2"
%L defabc        "1+Dy^2"
%L defabc        "2+Dy^2"
%L defabc     "Dx^2+Dy^2"
%L defabc   "1+Dx^2+Dy^2"
%L defabc   "2+Dx^2+Dy^2"
%L
%L defabc     "Dx^2-Dy^2"
%L defabc   "2+Dx^2-Dy^2"
%L
%L defabc     "Dx*Dy"
%L defabc   "2+Dx*Dy"
%L
%L defabc   "2*Dx^2+Dy^2"
%L defabc   "2*Dx^2-Dy^2"
\pu

\def\exprdxdyabc#1{\ensuremath{
  \ga{#1 1D} \quad
  \ga{#1 2D} \quad
  \ga{#1 3D}
  }}

\def\mabc#1{\ensuremath{
  \ga{#1 1D} &
  \scalebox{0.8}{\ga{#1 2D}} &
  \ga{#1 3D}
  }}

\newpage

% «alguns-exemplos»  (to ".alguns-exemplos")
% (c3m232mdp 3 "alguns-exemplos")
% (c3m232mda   "alguns-exemplos")
% (c3m232fhp 8 "alguns-exemplos")
% (c3m232fha   "alguns-exemplos")

{\bf Exemplos com numerozinhos}

\msk

\scalebox{0.6}{\def\colwidth{7cm}\firstcol{

$\begin{array}[t]{rcc}
 \mabc  {Dx^2}      \\
 \mabc       {Dy^2} \\
 \mabc  {Dx^2+Dy^2} \\ \\
 \mabc  {Dx^2-Dy^2} \\
 \mabc      {Dx*Dy} \\
 \end{array}
$

}\anothercol{

$\begin{array}[t]{rcc}
 \mabc      {2+Dx^2} \\
 \mabc      {2+Dy^2} \\
 \mabc {2+Dx^2+Dy^2} \\ \\
 \mabc {2+Dx^2-Dy^2} \\
 \mabc     {2+Dx*Dy} \\
 \end{array}
$

}}



\newpage

$\scalebox{1.6}{
 \begin{array}[t]{rcc}
 \mabc{2*Dx^2+Dy^2} \\
 \mabc{2*Dx^2-Dy^2} \\
 \end{array}
 }
$




\newpage

% «versao-mega-rapida»  (to ".versao-mega-rapida")
% (c3m232mdp 4 "versao-mega-rapida")
% (c3m232mda   "versao-mega-rapida")
% (c3m222mmsp 5 "versao-mega-rapida")
% (c3m222mmsa   "versao-mega-rapida")
% (find-angg ".emacs" "c3-2022-2")

{\bf Versão mega-rápida das páginas 365--394 do Bortolossi}


\scalebox{0.6}{\def\colwidth{15cm}\firstcol{

Links:

{\footnotesize

% (c3m222fhp 1 "title")
% (c3m222fha   "title")
%    http://angg.twu.net/LATEX/2022-2-C3-funcoes-homogeneas.pdf
\url{http://angg.twu.net/LATEX/2022-2-C3-funcoes-homogeneas.pdf}

% (find-angg ".emacs" "c3q222")
% (find-angg ".emacs" "c3q222" "Funções homogêneas")
%    http://angg.twu.net/2022.2-C3/C3-quadros.pdf#page=17
\url{http://angg.twu.net/2022.2-C3/C3-quadros.pdf\#page=17}

% (find-books "__analysis/__analysis.el" "bortolossi")
% http://angg.twu.net/2019.2-C3/Bortolossi/bortolossi-cap-10.pdf
% \url{http://angg.twu.net/2019.2-C3/Bortolossi/bortolossi-cap-10.pdf}

% (find-books "__analysis/__analysis.el" "bortolossi" "388" "positiva semidefinida")


}

\msk


Digamos que:

$$\begin{array}{rl}
  r_1,r_2,r∈\R, & r_1≠r_2, \\
  α∈\R, & α>0, \\
  β,γ∈\R, & γ>0, \\
  r_3=β+iγ, & r_4=β-iγ, \\
  z(x,y) \;\; = & dx^2 + exy + fy^2, \\
  h(x) \;\; = & z(x,1). \\
  \end{array}
$$

Então:

\msk

\begin{tabular}{lll}
se & $h(x) =   (x-r_1)(x-r_2)$ & então $(0,0)$ é um ponto de sela, \\
se & $h(x) =  α(x-r_1)(x-r_2)$ & então $(0,0)$ é um ponto de sela, \\
se & $h(x) = -α(x-r_1)(x-r_2)$ & então $(0,0)$ é um ponto de sela, \\
se & $h(x) =   (x-r)^2$ & então $(0,0)$ é como $z=x^2$, \\
se & $h(x) =  α(x-r)^2$ & então $(0,0)$ é como $z=x^2$, \\
se & $h(x) = -α(x-r)^2$ & então $(0,0)$ é como $z=-x^2$, \\
se & $h(x) =   (x-r_3)(x-r_4)$ & então $(0,0)$ ``tem concavidade pra cima'', \\
se & $h(x) =  α(x-r_3)(x-r_4)$ & então $(0,0)$ ``tem concavidade pra cima'', \\
se & $h(x) = -α(x-r_3)(x-r_4)$ & então $(0,0)$ ``tem concavidade pra baixo''. \\
\end{tabular}

}\anothercol{
}}



\newpage

\def\matz   {\pmat{z_{xx} & z_{xy} \\ z_{xy} & z_{yy}}}
\def\matabc {\pmat{2a & b \\ b & 2c}}
\def\matxtyt{\pmat{x_t \\ y_t}}
\def\matcs  {\pmat{c \\ s}}
\def\matmsc {\pmat{-s \\ c}}
\def\matLA  {\pmat{λ_1 & 0 \\ 0 & λ_2}}
\def\matxs  {\pmat{| & | \\ x_1 & x_2 \\ | & | }}
\def\matS   {\pmat{c & -s \\ s & c}}
\def\matSinv{\pmat{c & s \\ -s & c}}



\scalebox{0.55}{\def\colwidth{7.5cm}\firstcol{

$\begin{array}[t]{rl}
  \text{Se}    & z = z(x,y) = ax^2 + bxy + cy^2 \\
  \text{e}     & M = \matz \\
  \text{então} & z_{xx} = 2a \\
               & z_{xy} = b \\
               & z_{yy} = 2c \\
               & M = \matabc \\
  \end{array}
$

\bsk
\bsk

$\begin{array}[t]{rl}
  \text{Se}    & z = z(x,y) \\
               & x = x(t) \\
               & y = y(t) \\
               & x_{tt} = 0 \\
               & y_{tt} = 0 \\
  \text{então} & z_{tt} = z_{xx} x_t x_y + 2 z_{xy} x_t y_t + z_{yy} y_t y_t \\
  \text{e}     & z_{tt} = \matxtyt^T \matz \matxtyt \\
  \end{array}
$


}\def\colwidth{6cm}\anothercol{

$\begin{array}[t]{rl}
  \text{Se}    & c = \cosθ \\
               & s = \senθ \\
               & Λ = \matLA \\
               & x_1 = \matcs \\
               & x_2 = \matmsc \\
               & S = \matxs = \matS \\
               & A = SΛS^{-1} \\
  \text{então} & S^{-1} = \matSinv \\
               & S \pmat{1\\0} = x_1, \;\; S^{-1}x_1 = \pmat{1\\0} \\
               & S \pmat{0\\1} = x_2, \;\; S^{-1}x_2 = \pmat{0\\1} \\
               & Ax_1 = SΛS^{-1}x_1 = λ_1x_1 \\
               & Ax_2 = SΛS^{-1}x_2 = λ_2x_2 \\
               & x_1^T A x_1 = λ_1 \\
               & x_2^T A x_2 = λ_2 \\
  \end{array}
$

}\anothercol{


$\begin{array}[t]{rl}
  \text{Se além disso} \\
  \text{tudo temos} & M = A = SΛS^1 \\
  \text{então quando} & \matxtyt = x_1 = \matcs \\
  \text{temos}        & z_{tt} = λ_1 \\
  \text{e quando}     & \matxtyt = x_2 = \matmsc \\
  \text{temos}        & z_{tt} = λ_2. \\
  \end{array}
$



}}


\newpage

% «eigshows»  (to ".eigshows")
% 3hT133 (c3m232mdp 7 "eigshows")
%        (c3m232mda   "eigshows")
%        (find-angg "MAXIMA/eigshow1.mac")
% (find-fline "~/LATEX/2023-2-C3/")

\def\eigshowabn#1{
  \hspace*{-1cm}
  \includegraphics[height=6cm]{2023-2-C3/eigshow_a_#1.pdf}
  \hspace*{-1cm}
  \includegraphics[height=6cm]{2023-2-C3/eigshow_b_#1.pdf}
}

\eigshowabn{0} \newpage
\eigshowabn{1} \newpage
\eigshowabn{2} \newpage
\eigshowabn{3} \newpage
\eigshowabn{4} \newpage
\eigshowabn{5} \newpage
\eigshowabn{6} \newpage
\eigshowabn{7} \newpage
\eigshowabn{8} \newpage
\eigshowabn{9} \newpage
\eigshowabn{10} \newpage
\eigshowabn{11} \newpage
\eigshowabn{12} \newpage
\eigshowabn{13} \newpage
\eigshowabn{14} \newpage
\eigshowabn{15} \newpage
\eigshowabn{16} \newpage




\GenericWarning{Success:}{Success!!!}  % Used by `M-x cv'

\end{document}

%  ____  _             _         
% |  _ \(_)_   ___   _(_)_______ 
% | | | | \ \ / / | | | |_  / _ \
% | |_| | |\ V /| |_| | |/ /  __/
% |____// | \_/  \__,_|_/___\___|
%     |__/                       
%
% «djvuize»  (to ".djvuize")
% (find-LATEXgrep "grep --color -nH --null -e djvuize 2020-1*.tex")

 (eepitch-shell)
 (eepitch-kill)
 (eepitch-shell)
# (find-fline "~/2023.2-C3/")
# (find-fline "~/LATEX/2023-2-C3/")
# (find-fline "~/bin/djvuize")

cd /tmp/
for i in *.jpg; do echo f $(basename $i .jpg); done

f () { rm -v $1.pdf;  textcleaner -f 50 -o  5 $1.jpg $1.png; djvuize $1.pdf; xpdf $1.pdf }
f () { rm -v $1.pdf;  textcleaner -f 50 -o 10 $1.jpg $1.png; djvuize $1.pdf; xpdf $1.pdf }
f () { rm -v $1.pdf;  textcleaner -f 50 -o 20 $1.jpg $1.png; djvuize $1.pdf; xpdf $1.pdf }

f () { rm -fv $1.png $1.pdf; djvuize $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 1.0 -f 15" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 1.0 -f 30" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 1.0 -f 45" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 0.5" $1.pdf; xpdf $1.pdf }
f () { rm -fv $1.png $1.pdf; djvuize WHITEBOARDOPTS="-m 0.25" $1.pdf; xpdf $1.pdf }
f () { cp -fv $1.png $1.pdf       ~/2023.2-C3/
       cp -fv        $1.pdf ~/LATEX/2023-2-C3/
       cat <<%%%
% (find-latexscan-links "C3" "$1")
%%%
}

f 20201213_area_em_funcao_de_theta
f 20201213_area_em_funcao_de_x
f 20201213_area_fatias_pizza



%  __  __       _        
% |  \/  | __ _| | _____ 
% | |\/| |/ _` | |/ / _ \
% | |  | | (_| |   <  __/
% |_|  |_|\__,_|_|\_\___|
%                        
% <make>

 (eepitch-shell)
 (eepitch-kill)
 (eepitch-shell)
# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2023-2-C3-matrizes-definidas veryclean
make -f 2019.mk STEM=2023-2-C3-matrizes-definidas pdf

% Local Variables:
% coding: utf-8-unix
% ee-tla: "c3md"
% ee-tla: "c3m232md"
% End:
