diff options
| -rw-r--r-- | anm/n.lyx | 17 | ||||
| -rw-r--r-- | anm/n2.lyx | 20 | ||||
| -rw-r--r-- | anm/n4.lyx | 2151 |
3 files changed, 2184 insertions, 4 deletions
@@ -5,6 +5,9 @@ \save_transient_properties true \origin unavailable \textclass book +\begin_preamble +\usepackage{blkarray} +\end_preamble \use_default_options true \begin_modules algorithm2e @@ -210,6 +213,20 @@ filename "n3.lyx" \end_layout \begin_layout Chapter +Valores y vectores propios +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand input +filename "n4.lyx" + +\end_inset + + +\end_layout + +\begin_layout Chapter \start_of_appendix Octave \end_layout @@ -1140,11 +1140,15 @@ En el algoritmo de Gauss sin permutaciones de filas, se inicializa \begin_inset Formula $A^{(k)}$ \end_inset - como en + como en las primeras +\begin_inset Formula $k-1$ +\end_inset + + columnas de \begin_inset Formula $L$ \end_inset - y + y en \begin_inset Formula $P$ \end_inset @@ -1216,11 +1220,19 @@ En el algoritmo de Gauss sin permutaciones de filas, se inicializan \begin_inset Formula $A^{(k)}$ \end_inset - y +, \begin_inset Formula $P$ \end_inset - y las columnas + y las primeras +\begin_inset Formula $k-1$ +\end_inset + + columnas de +\begin_inset Formula $L$ +\end_inset + +, y las columnas \begin_inset Formula $j$ \end_inset diff --git a/anm/n4.lyx b/anm/n4.lyx new file mode 100644 index 0000000..7649537 --- /dev/null +++ b/anm/n4.lyx @@ -0,0 +1,2151 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass book +\begin_preamble +\input{../defs} +\usepackage{blkarray} +\end_preamble +\use_default_options true +\begin_modules +algorithm2e +\end_modules +\maintain_unincluded_children false +\language spanish +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style french +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Standard + +\series bold +Teorema de los círculos de Gershgorin: +\series default + El conjunto de valores propios de +\begin_inset Formula $A\in{\cal M}_{n}(\mathbb{C})$ +\end_inset + + está contenido en +\begin_inset Formula +\[ +\bigcup_{k=1}^{n}\overline{B}\left(a_{kk},\sum_{\begin{subarray}{c} +j=1\\ +j\neq k +\end{subarray}}^{n}|a_{kj}|\right). +\] + +\end_inset + + +\series bold +Demostración: +\series default + Si +\begin_inset Formula $\lambda$ +\end_inset + + no está contenido en este conjunto, para cada +\begin_inset Formula $k$ +\end_inset + +, +\begin_inset Formula $|a_{kk}-\lambda|>\sum_{j\neq k}|a_{kj}|$ +\end_inset + +, luego +\begin_inset Formula $A-\lambda I$ +\end_inset + + tiene diagonal estrictamente dominante y por tanto es no singular y +\begin_inset Formula $\lambda$ +\end_inset + + no es valor propio de +\begin_inset Formula $A$ +\end_inset + +. +\end_layout + +\begin_layout Section +Método de la potencia o del cociente de Rayleigh +\end_layout + +\begin_layout Standard +Sean +\begin_inset Formula $A\in{\cal M}_{n}(\mathbb{C})$ +\end_inset + + con valores propios +\begin_inset Formula $\lambda_{1},\dots,\lambda_{n}$ +\end_inset + + dispuestos tal que +\begin_inset Formula $|\lambda_{1}|\geq\dots\geq|\lambda_{n}|$ +\end_inset + + y vectores propios respectivos +\begin_inset Formula $v_{1},\dots,v_{n}$ +\end_inset + + formando una base ortogonal de +\begin_inset Formula $\mathbb{C}^{n}$ +\end_inset + +, +\begin_inset Formula $p,y\in\mathbb{C}$ +\end_inset + +, si +\begin_inset Formula $\langle x_{0},v_{1}\rangle\neq0$ +\end_inset + + y +\begin_inset Formula $\langle v_{1},y\rangle\neq0$ +\end_inset + +, las sucesiones +\begin_inset Formula $(x_{k})_{k}$ +\end_inset + + y +\begin_inset Formula $(r_{k})_{k}$ +\end_inset + + dadas por +\begin_inset Formula $x_{0}:=p$ +\end_inset + +, +\begin_inset Formula $x_{k+1}:=Ax_{k}$ +\end_inset + + y +\begin_inset Formula $r_{k}=\frac{\langle x_{k+1},y\rangle}{\langle x_{k},y\rangle}$ +\end_inset + +, entonces +\begin_inset Formula $(r_{k})_{k}$ +\end_inset + + está bien definida y converge a +\begin_inset Formula $\lambda_{1}$ +\end_inset + +, y +\begin_inset Formula $\frac{x_{2k}}{\Vert x_{2k}\Vert}$ +\end_inset + + converge a un múltiplo de +\begin_inset Formula $v_{1}$ +\end_inset + +. + +\series bold +Demostración: +\series default + Sean +\begin_inset Formula $\phi(x):=\langle x,y\rangle$ +\end_inset + +, +\begin_inset Formula $p=:\alpha_{1}v_{1}+\dots+\alpha_{n}v_{n}$ +\end_inset + +, se tiene +\begin_inset Formula $x_{k}=A^{k}p$ +\end_inset + +, con lo que suponiendo +\begin_inset Formula $|\lambda_{1}|>|\lambda_{2}|$ +\end_inset + +, +\begin_inset Formula +\[ +x_{k}=\alpha_{1}\lambda_{1}^{k}v_{1}+\dots+\alpha_{n}\lambda_{n}^{k}v_{n}=\lambda_{1}^{k}\left(\alpha_{1}v_{1}+\sum_{j=2}^{n}\left(\frac{\lambda_{j}}{\lambda_{1}}\right)^{k}\alpha_{j}v_{j}\right)=:\lambda_{1}^{k}(\alpha_{1}v_{1}+\varepsilon_{k}). +\] + +\end_inset + +Es claro que +\begin_inset Formula $\varepsilon_{k}\to0$ +\end_inset + +, luego +\begin_inset Formula $\lim_{2k}\frac{x_{2k}}{\Vert x_{2k}\Vert}=\lim_{k}\frac{\alpha_{1}v_{1}+\varepsilon_{2k}}{\Vert\alpha_{1}v_{1}+\varepsilon_{2k}\Vert}=\frac{\alpha_{1}v_{1}}{\Vert\alpha_{1}v_{1}\Vert}$ +\end_inset + + y por ser +\begin_inset Formula $\phi$ +\end_inset + + lineal, como +\begin_inset Formula $\alpha_{1}\neq0$ +\end_inset + + y +\begin_inset Formula $\phi(v_{1})\neq0$ +\end_inset + +, +\begin_inset Formula +\[ +\lim_{k}r_{k}=\lim_{k}\frac{\phi(x_{k+1})}{\phi(x_{k})}=\lim_{k}\frac{\lambda_{1}^{k+1}\phi(\alpha_{1}v_{1}+\varepsilon_{k+1})}{\lambda_{1}^{k}\phi(\alpha_{1}v_{1}+\varepsilon_{k})}=\lambda_{1}\lim_{k}\frac{\alpha_{1}\phi(v_{1})+\phi(\varepsilon_{k+1})}{\alpha_{1}\phi(v_{1})+\phi(\varepsilon_{k+1})}=\lambda_{1}. +\] + +\end_inset + +Es fácil ver cómo se generalizaría esto para cuando los +\begin_inset Formula $j\in\{1,\dots,n\}$ +\end_inset + + primeros valores propios tienen igual valor absoluto. +\end_layout + +\begin_layout Standard +El +\series bold +método de la potencia +\series default + o +\series bold +del cociente de Rayleigh +\series default + consiste en tomar +\begin_inset Formula $p,y\in\mathbb{C}$ +\end_inset + + arbitrarios en lo anterior, pues todavía no conocemos +\begin_inset Formula $v_{1}$ +\end_inset + +, e ir construyendo +\begin_inset Formula $(x_{k})_{k}$ +\end_inset + + y +\begin_inset Formula $(r_{k})_{k}$ +\end_inset + + para obtener el valor propio de +\begin_inset Formula $A$ +\end_inset + + con mayor valor absoluto. + +\end_layout + +\begin_layout Standard +En la práctica no se calcula +\begin_inset Formula $(x_{k})_{k}$ +\end_inset + + directamente, pues puede tender a infinito o cero y esto da errores de + condicionamiento. + En su lugar se calcula +\begin_inset Formula $(y_{k})_{k}$ +\end_inset + + dada por +\begin_inset Formula $y_{0}:=\frac{x_{0}}{\Vert x_{0}\Vert}$ +\end_inset + + e +\begin_inset Formula $y_{k+1}:=\frac{Ay_{k}}{\Vert Ay_{k}\Vert}$ +\end_inset + +, y entonces +\begin_inset Formula $r_{k}=\frac{\langle Ay_{k},y\rangle}{\langle y_{k},y\rangle}$ +\end_inset + +. + En efecto, si +\begin_inset Formula $y_{k}=\frac{x_{k}}{\Vert x_{k}\Vert}$ +\end_inset + +, +\begin_inset Formula $y_{k+1}=\frac{Ay_{k}}{\Vert Ay_{k}\Vert}=\frac{Ax_{k}}{\Vert Ax_{k}\Vert}=\frac{x_{k+1}}{\Vert x_{k+1}\Vert}$ +\end_inset + +, luego por inducción esto ocurre para todo +\begin_inset Formula $k$ +\end_inset + +, y entonces, como +\begin_inset Formula $\Vert x_{k+1}\Vert=\Vert Ax_{k}\Vert=\Vert Ay_{k}\Vert\Vert x_{k}\Vert$ +\end_inset + +, +\begin_inset Formula +\[ +r_{k}=\frac{\langle x_{k+1},y\rangle}{\langle x_{k},y\rangle}=\frac{\Vert x_{k+1}\Vert\langle y_{k+1},y\rangle}{\Vert x_{k}\Vert\langle y_{k},y\rangle}=\frac{\Vert x_{k+1}\Vert\langle Ay_{k},y\rangle}{\Vert x_{k}\Vert\Vert Ay_{k}\Vert\langle y_{k},y\rangle}=\frac{\langle Ay_{k},y\rangle}{\langle y_{k},y\rangle}. +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $A$ +\end_inset + + es invertible, el +\series bold +método de la potencia inversa +\series default + consiste en aplicar el método de la potencia a +\begin_inset Formula $A^{-1}$ +\end_inset + +, obteniendo el inverso del valor propio de +\begin_inset Formula $A$ +\end_inset + + con menor valor absoluto, pues +\begin_inset Formula $Au=\lambda u\iff\lambda^{-1}u=A^{-1}u$ +\end_inset + +. + Para ello, se factoriza +\begin_inset Formula $A$ +\end_inset + + y bien se obtiene +\begin_inset Formula $A^{-1}$ +\end_inset + + resolviendo columna a columna o se resuelve en cada paso +\begin_inset Formula $Ax_{k+1}=x_{k}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Los valores propios de +\begin_inset Formula $A-\mu I$ +\end_inset + + son de la forma +\begin_inset Formula $\lambda-\mu$ +\end_inset + +, siendo +\begin_inset Formula $\lambda$ +\end_inset + + un valor propio de +\begin_inset Formula $A$ +\end_inset + +, por lo que los métodos de la potencia y la potencia inversa sobre +\begin_inset Formula $A-\mu I$ +\end_inset + +, llamados de la potencia y la potencia inversa +\series bold +con desplazamiento +\series default +, nos darían respectivamente el valor propio más lejano y más cercano a + +\begin_inset Formula $\mu$ +\end_inset + +. +\end_layout + +\begin_layout Section +Método de Jacobi +\end_layout + +\begin_layout Standard +Sea +\begin_inset Formula $A\in{\cal M}_{n}(\mathbb{R})$ +\end_inset + + simétrica, y por tanto diagonalizable. + Entonces el problema de encontrar los valores y vectores propios de una + matriz se puede traducir en el de encontrar una matriz ortogonal que diagonalic +e +\begin_inset Formula $A$ +\end_inset + +, que en el caso de +\begin_inset Formula $n=2$ +\end_inset + + será un giro. + El +\series bold +método de Jacobi +\series default + consiste en construir una sucesión +\begin_inset Formula $(O_{k})_{k}$ +\end_inset + + de giros en planos determinados por dos vectores de la base canónica de + forma que +\begin_inset Formula $(A_{k}:=(O_{1}\cdots O_{k})^{t}A(O_{1}\cdots O_{k}))_{k}$ +\end_inset + +, que podemos obtener como +\begin_inset Formula $A_{0}=A$ +\end_inset + + y +\begin_inset Formula $A_{k+1}=O_{k+1}^{t}A_{k}O_{k+1}$ +\end_inset + +, converja a una matriz diagonal. +\end_layout + +\begin_layout Standard +Sean +\begin_inset Formula $1\leq p<q\leq n$ +\end_inset + +, +\begin_inset Formula $\theta\in\mathbb{R}$ +\end_inset + +, +\begin_inset Formula $A:=(a_{ij})\in{\cal M}_{n}(\mathbb{R})$ +\end_inset + + simétrica, +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +bgroup +\backslash +small +\backslash +[ +\end_layout + +\begin_layout Plain Layout + +O:= +\backslash +begin{blockarray}{cccccccccccc} +\end_layout + +\begin_layout Plain Layout + + &&&p&&&&q&&&& +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + + &&& +\backslash +downarrow&&&& +\backslash +downarrow&&&& +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + + +\backslash +begin{block}{(ccccccccccc)c} +\end_layout + +\begin_layout Plain Layout + + 1 +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + + & +\backslash +ddots +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + + &&1 +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + + &&& +\backslash +cos +\backslash +theta&&&& +\backslash +sin +\backslash +theta&&&& +\backslash +gets p +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + + &&&&1 +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + + &&&&& +\backslash +ddots +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + + &&&&&&1 +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + + &&&- +\backslash +sin +\backslash +theta&&&& +\backslash +cos +\backslash +theta&&&& +\backslash +gets q +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + + &&&&&&&&1 +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + + &&&&&&&&& +\backslash +ddots +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + + &&&&&&&&&&1 +\backslash + +\backslash + +\end_layout + +\begin_layout Plain Layout + + +\backslash +end{block} +\end_layout + +\begin_layout Plain Layout + + +\backslash +end{blockarray}, +\end_layout + +\begin_layout Plain Layout + + +\backslash +] +\backslash +egroup +\end_layout + +\end_inset + +y +\begin_inset Formula $B:=(b_{ij}):=O^{t}AO$ +\end_inset + +, entonces: +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $B$ +\end_inset + + es simétrica y cumple +\begin_inset Formula $\sum_{i,j}b_{ij}^{2}=\sum_{i,j}a_{ij}^{2}$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +\begin_inset Formula $B^{t}=(O^{t}AO)^{t}=O^{t}A^{t}O=O^{t}AO=B$ +\end_inset + +, luego +\begin_inset Formula $B$ +\end_inset + + es simétrica. + +\begin_inset Formula $B^{t}B=O^{t}A^{t}OO^{t}AO=O^{t}A^{t}AO$ +\end_inset + +, luego +\begin_inset Formula $\text{tr}(B^{t}B)=\text{tr}(A^{t}A)$ +\end_inset + +, pero +\begin_inset Formula +\[ +\text{tr}(A^{t}A)=\sum_{j}(A^{t}A)_{j}=\sum_{j,i}(A^{*})_{ji}A_{ij}=\sum_{i,j}a_{ij}^{2}, +\] + +\end_inset + +y análogamente, +\begin_inset Formula $\text{tr}(B^{t}B)=\sum_{i,j}b_{ij}^{2}$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +Si +\begin_inset Formula $a_{pq}\neq0$ +\end_inset + +, +\begin_inset Formula $b_{pq}=0$ +\end_inset + + si y sólo si +\begin_inset Formula $\cot(2\theta)=\frac{a_{qq}-a_{pp}}{2a_{pq}}$ +\end_inset + +, con lo que el valor de +\begin_inset Formula $\theta\in(-\frac{\pi}{4},\frac{\pi}{4}]\setminus\{0\}$ +\end_inset + + que cumple esto es único, y para dicho valor, +\begin_inset Formula $\sum_{k}b_{kk}^{2}=\sum_{k}a_{kk}^{2}+2a_{pq}^{2}$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Como +\begin_inset Formula +\[ +\left(\begin{array}{cc} +b_{pp} & b_{pq}\\ +b_{qp} & b_{qq} +\end{array}\right)=\left(\begin{array}{cc} +\cos\theta & -\sin\theta\\ +\sin\theta & \cos\theta +\end{array}\right)\left(\begin{array}{cc} +a_{pp} & a_{pq}\\ +a_{qp} & a_{qq} +\end{array}\right)\left(\begin{array}{cc} +\cos\theta & \sin\theta\\ +-\sin\theta & \cos\theta +\end{array}\right), +\] + +\end_inset + + +\begin_inset Formula +\begin{align*} +b_{pq} & =\cos\theta(a_{pp}\sin\theta+a_{pq}\cos\theta)-\sin\theta(a_{qp}\sin\theta+a_{qq}\cos\theta)\\ + & =(a_{pp}-a_{qq})\sin\theta\cos\theta+a_{pq}(\cos^{2}\theta-\sin^{2}\theta)\\ + & =\frac{a_{pp}-a_{qq}}{2}\sin(2\theta)+a_{pq}\cos(2\theta), +\end{align*} + +\end_inset + +de donde se obtiene la primera parte del enunciado. + Aplicando el punto 1 a las submatrices cuadradas de +\begin_inset Formula $A$ +\end_inset + + y +\begin_inset Formula $B$ +\end_inset + + con las filas y columnas +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $q$ +\end_inset + +, +\begin_inset Formula $a_{pp}^{2}+a_{qq}^{2}+2a_{pq}^{2}=b_{pp}^{2}+b_{qq}^{2}+2b_{pq}^{2}$ +\end_inset + +. + Por la estructura de +\begin_inset Formula $O$ +\end_inset + +, las columnas de +\begin_inset Formula $AO$ +\end_inset + + son las de +\begin_inset Formula $A$ +\end_inset + + excepto +\begin_inset Formula $A_{p}$ +\end_inset + + y +\begin_inset Formula $A_{q}$ +\end_inset + +, y dada +\begin_inset Formula $C:=(c_{ij})\in{\cal M}_{n}(\mathbb{R})$ +\end_inset + +, +\begin_inset Formula $O^{t}C$ +\end_inset + + tiene las mismas filas que +\begin_inset Formula $C$ +\end_inset + + salvo +\begin_inset Formula $c_{p}$ +\end_inset + + y +\begin_inset Formula $c_{q}$ +\end_inset + +. + Entonces +\begin_inset Formula $b_{kk}=a_{kk}$ +\end_inset + + para +\begin_inset Formula $k\neq i,j$ +\end_inset + + pero +\begin_inset Formula $b_{pq}^{2}=0$ +\end_inset + +, luego +\begin_inset Formula $\sum_{k}b_{kk}^{2}=\sum_{k\neq i,j}b_{kk}^{2}+b_{pp}^{2}+b_{qq}^{2}=\sum_{k\neq i,j}a_{kk}^{2}+a_{pp}^{2}+a_{qq}^{2}+2a_{pq}^{2}=\sum_{k}a_{kk}^{2}+2a_{pq}^{2}$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +Para el +\begin_inset Formula $\theta$ +\end_inset + + descrito en el apartado anterior, sean +\begin_inset Formula $x:=\frac{a_{qq}-a_{pp}}{2a_{pq}}$ +\end_inset + +, +\begin_inset Formula +\[ +t:=\begin{cases} +-x+\sqrt{x^{2}+1} & \text{si }x\geq0,\\ +-x-\sqrt{x^{2}+1} & \text{si }x<0, +\end{cases} +\] + +\end_inset + + +\begin_inset Formula $c:=\frac{1}{\sqrt{1+t^{2}}}$ +\end_inset + + y +\begin_inset Formula $s:=\frac{t}{\sqrt{1+t^{2}}}$ +\end_inset + +, para +\begin_inset Formula $i,j\neq p,q$ +\end_inset + +, +\begin_inset Formula +\begin{align*} +b_{pp} & =a_{pp}-ta_{pq}, & b_{qq} & =a_{qq}+ta_{pq}, & b_{pq} & =0,\\ +b_{pi}=b_{ip} & =ca_{ip}-sa_{iq}, & b_{qi}=b_{iq} & =sa_{ip}+ca_{iq}, & b_{ij} & =a_{ij}. +\end{align*} + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout Standard +Sean +\begin_inset Formula $x:=\frac{a_{qq}-a_{pp}}{2a_{pq}}$ +\end_inset + + y +\begin_inset Formula $t:=\tan\theta$ +\end_inset + +. + Entonces +\begin_inset Formula $x=\cot2\theta=\frac{\cos^{2}\theta-\sin^{2}\theta}{2\sin\theta\cos\theta}=\frac{1-\tan^{2}\theta}{2\tan\theta}=\frac{1-t^{2}}{2t}$ +\end_inset + +, luego +\begin_inset Formula $t^{2}-2xt-1=0$ +\end_inset + + y +\begin_inset Formula $t=\frac{2x\pm\sqrt{4x^{2}+4}}{2}=x\pm\sqrt{x^{2}+1}$ +\end_inset + +, y como +\begin_inset Formula $|t|\leq1$ +\end_inset + + porque +\begin_inset Formula $|\theta|\leq\frac{\pi}{4}$ +\end_inset + +, queda el valor de +\begin_inset Formula $t$ +\end_inset + + dado. + Como +\begin_inset Formula $|\theta|\leq\frac{\pi}{4}$ +\end_inset + +, +\begin_inset Formula $\cos\theta>0$ +\end_inset + +, y como +\begin_inset Formula $\tan^{2}\theta+1=\frac{1}{\cos^{2}\theta}$ +\end_inset + +, +\begin_inset Formula $\cos\theta=c$ +\end_inset + + y +\begin_inset Formula $\sin\theta=s$ +\end_inset + +. + Entonces los casos +\begin_inset Formula $b_{pi}$ +\end_inset + +, +\begin_inset Formula $b_{qi}$ +\end_inset + +, +\begin_inset Formula $b_{ip}$ +\end_inset + + y +\begin_inset Formula $b_{iq}$ +\end_inset + + son obvios, y +\begin_inset Formula $b_{pq}$ +\end_inset + + y +\begin_inset Formula $b_{qp}$ +\end_inset + + vienen dados por el ejercicio anterior. + Finalmente, +\begin_inset Formula +\begin{align*} +b_{pp} & =c(a_{pp}c-a_{qp}s)-s(a_{qp}c-a_{qq}s)=a_{pp}c^{2}+a_{qq}s^{2}-2csa_{pq}=\\ + & =a_{pp}+s^{2}(a_{qq}-a_{pp})-2csa_{pq}=a_{pp}+\frac{t^{2}}{t^{2}+1}x2a_{pq}-\frac{2t}{t^{2}+1}a_{pq}=\\ + & =a_{pp}+\frac{t(1-t^{2})}{t^{2}+1}a_{pq}-\frac{2t}{t^{2}+1}a_{pq}=a_{pp}-ta_{pq};\\ +b_{qq} & =s(a_{pp}s+a_{pq}c)+c(a_{qp}s+a_{qq}c)=a_{pp}s^{2}+a_{qq}c^{2}+2csa_{pq}=\\ + & =a_{qq}+s^{2}(a_{pp}-a_{qq})+2csa_{pq}=a_{qq}-\frac{t^{2}}{t^{2}+1}x2a_{pq}+\frac{2t}{t^{2}+1}a_{pq}=\\ + & =a_{qq}-\frac{t(1-t^{2})}{t^{2}+1}a_{pq}+\frac{2t}{t^{2}+1}a_{pq}=a_{qq}+ta_{pq}. +\end{align*} + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Standard +\begin_inset Float algorithm +wide false +sideways false +status open + +\begin_layout Plain Layout +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +Entrada{Matriz simétrica real $A:=(a_{ij})$ de tamaño $n$ y nivel de tolerancia + a errores $e>0$.} +\end_layout + +\begin_layout Plain Layout + + +\backslash +Salida{Vector $ +\backslash +lambda$ de tamaño $n$ que aproxima los valores propios de $A$ y matriz ortogonal + $ +\backslash +Omega$ de tamaño $n$ cuyas columnas aproximan los correspondientes vectores + propios.} +\end_layout + +\begin_layout Plain Layout + +$ +\backslash +Omega +\backslash +gets I_n$ +\backslash +; +\end_layout + +\begin_layout Plain Layout + + +\backslash +Mientras{$ +\backslash +sum_{1 +\backslash +leq i<j +\backslash +leq n} a_{ij} +\backslash +leq e$}{ +\end_layout + +\begin_layout Plain Layout + + +\backslash +tcp{{ +\backslash +rm Elegimos $p$ y $q$ por el { +\backslash +bf criterio de Jacobi clásico}, y +\end_layout + +\begin_layout Plain Layout + + por la condición de parada elegida, $a_{pq}>0$.}} +\end_layout + +\begin_layout Plain Layout + + Establecer $p<q$ tales que $|a_{pq}|= +\backslash +max_{i<j}|a_{ij}|$ +\backslash +; +\end_layout + +\begin_layout Plain Layout + + $x +\backslash +gets{a_{qq}-a{pp} +\backslash +over2a_{pq}}$ +\backslash +; +\end_layout + +\begin_layout Plain Layout + + +\backslash +lSSi{$x +\backslash +geq0$}{$t +\backslash +gets-x+ +\backslash +sqrt{x^2+1}$} +\end_layout + +\begin_layout Plain Layout + + +\backslash +lEnOtroCaso{$t +\backslash +gets-x- +\backslash +sqrt{x^2+1}$} +\end_layout + +\begin_layout Plain Layout + + $c +\backslash +gets{1 +\backslash +over +\backslash +sqrt{1+t^2}}$ +\backslash +; +\end_layout + +\begin_layout Plain Layout + + $s +\backslash +gets{t +\backslash +over +\backslash +sqrt{1+t^2}}$ +\backslash +; +\end_layout + +\begin_layout Plain Layout + + $b_{pp} +\backslash +gets a_{pp}-ta_{pq}$ +\backslash +; +\end_layout + +\begin_layout Plain Layout + + $b_{qq} +\backslash +gets a_{qq}+ta_{pq}$ +\backslash +; +\end_layout + +\begin_layout Plain Layout + + $b_{pq},b_{qp} +\backslash +gets0$ +\backslash +; +\end_layout + +\begin_layout Plain Layout + + +\backslash +Para{$i +\backslash +neq p,q$}{ +\end_layout + +\begin_layout Plain Layout + + $b_{pi},b_{ip} +\backslash +gets ca_{ip}-sa_{iq}$ +\backslash +; +\end_layout + +\begin_layout Plain Layout + + $b_{qi},b_{iq} +\backslash +gets sa_{ip}+ca_{iq}$ +\backslash +; +\end_layout + +\begin_layout Plain Layout + + } +\end_layout + +\begin_layout Plain Layout + + +\backslash +lPara{$i,j +\backslash +neq p,q$}{$b_{ij} +\backslash +gets a_{ij}$} +\end_layout + +\begin_layout Plain Layout + + $A +\backslash +gets (b_{ij})$ +\end_layout + +\begin_layout Plain Layout + + +\backslash +tcp*{$A +\backslash +gets O^tAO$} +\end_layout + +\begin_layout Plain Layout + + +\backslash +Para{$i +\backslash +gets1$ +\backslash +KwA $n$}{ +\end_layout + +\begin_layout Plain Layout + + $o_{ip} +\backslash +gets c +\backslash +omega_{ip}-s +\backslash +omega_{iq}$ +\backslash +; +\end_layout + +\begin_layout Plain Layout + + $o_{iq} +\backslash +gets s +\backslash +omega_{ip}-c +\backslash +omega_{iq}$ +\backslash +; +\end_layout + +\begin_layout Plain Layout + + +\backslash +lPara{$j +\backslash +neq p,q$}{$o_{ij} +\backslash +gets +\backslash +omega_{ij}$} +\end_layout + +\begin_layout Plain Layout + + } +\end_layout + +\begin_layout Plain Layout + + $ +\backslash +Omega +\backslash +gets (o_{ij})$ +\end_layout + +\begin_layout Plain Layout + + +\backslash +tcp*{$ +\backslash +Omega +\backslash +gets +\backslash +Omega*O$} +\end_layout + +\begin_layout Plain Layout + +} +\end_layout + +\begin_layout Plain Layout + +$ +\backslash +lambda +\backslash +gets +\backslash +text{diagonal de }A$ +\backslash +; +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "alg:jacobi" + +\end_inset + +Método de Jacobi clásico. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Con esto, el método de Jacobi clásico es el algoritmo +\begin_inset CommandInset ref +LatexCommand ref +reference "alg:jacobi" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, que en cada iteración multiplica implícitamente +\begin_inset Formula $A$ +\end_inset + + por la matriz de giro +\begin_inset Formula $O$ +\end_inset + + que anula +\begin_inset Formula $a_{pq}$ +\end_inset + +. + +\end_layout + +\begin_layout Standard +El +\series bold +teorema de convergencia del método de Jacobi clásico +\series default + nos dice que, dada una matriz +\begin_inset Formula $A\in{\cal M}_{n}(\mathbb{R})$ +\end_inset + + simétrica, si para +\begin_inset Formula $k\geq0$ +\end_inset + + llamamos +\begin_inset Formula $A_{k}$ +\end_inset + + a la matriz +\begin_inset Formula $A$ +\end_inset + + tras +\begin_inset Formula $k$ +\end_inset + + iteraciones del bucle y +\begin_inset Formula $\Omega_{k}$ +\end_inset + + a la matriz +\begin_inset Formula $\Omega$ +\end_inset + + tras +\begin_inset Formula $k$ +\end_inset + + iteraciones, ignorando la condición de parada y dejando +\begin_inset Formula $A$ +\end_inset + + y +\begin_inset Formula $\Omega$ +\end_inset + + sin modificar si +\begin_inset Formula $A$ +\end_inset + + es diagonal, +\begin_inset Formula $(A_{k})_{k}$ +\end_inset + + converge a una matriz diagonal cuya diagonal está formada por los valores + propios de +\begin_inset Formula $A$ +\end_inset + +, y si además estos son distintos dos a dos, +\begin_inset Formula $(\Omega_{k})_{k}$ +\end_inset + + converge a una matriz ortogonal cuyas columnas son los correspondientes + vectores propios de +\begin_inset Formula $A$ +\end_inset + +, en el mismo orden. +\end_layout + +\begin_layout Standard + +\series bold +Demostración: +\series default + Si algún +\begin_inset Formula $A_{k}$ +\end_inset + + es diagonal, esto ya ocurre, pues +\begin_inset Formula $\Omega_{k}$ +\end_inset + + es ortogonal y +\begin_inset Formula $A_{k}=\Omega_{k}^{t}A\Omega_{k}$ +\end_inset + +, por lo que supondremos que ningún +\begin_inset Formula $A_{k}$ +\end_inset + + lo es. + Por tanto +\begin_inset Formula $A$ +\end_inset + + es de tamaño +\begin_inset Formula $n\geq3$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Vemos primero que, si +\begin_inset Formula $(x_{k})_{k}$ +\end_inset + + es una sucesión acotada en un +\begin_inset Formula $\mathbb{R}$ +\end_inset + +-espacio normado +\begin_inset Formula $X$ +\end_inset + + de dimensión finita con una cantidad finita de puntos de acumulación +\begin_inset Formula $a_{1},\dots,a_{M}$ +\end_inset + + y +\begin_inset Formula +\[ +\lim_{k}\Vert x_{k+1}-x_{k}\Vert=0, +\] + +\end_inset + + entonces +\begin_inset Formula $(x_{k})_{k}$ +\end_inset + + es convergente. + Sea +\begin_inset Formula +\[ +\epsilon:=\frac{1}{3}\min_{i\neq j}\Vert a_{i}-a_{j}\Vert>0. +\] + +\end_inset + +Existe un +\begin_inset Formula $k_{0}\in\mathbb{N}$ +\end_inset + + tal que para +\begin_inset Formula $k\geq k_{0}$ +\end_inset + +, +\begin_inset Formula +\[ +x_{k}\in\bigcup_{k=1}^{M}B(a_{k},\epsilon), +\] + +\end_inset + +pues en otro caso existiría una subsucesión +\begin_inset Formula $(x_{k_{m}})_{m}$ +\end_inset + + de +\begin_inset Formula $(x_{k})_{k}$ +\end_inset + + tal que +\begin_inset Formula $x_{k_{m}}\notin\bigcup_{i=1}^{M}B(a_{k},\epsilon)$ +\end_inset + +, pero esta subsucesión está en un espacio acotado y, por tanto, tiene un + punto de acumulación. +\begin_inset Formula $\#$ +\end_inset + + +\end_layout + +\begin_layout Standard +Como +\begin_inset Formula $\lim_{k}\Vert x_{k+1}-x_{k}\Vert=0$ +\end_inset + +, existe +\begin_inset Formula $k_{1}\geq k_{0}$ +\end_inset + + tal que para +\begin_inset Formula $k\geq k_{1}$ +\end_inset + +, +\begin_inset Formula $\Vert x_{k+1}-x_{k}\Vert<\epsilon$ +\end_inset + +. + Sea +\begin_inset Formula $i_{0}$ +\end_inset + + tal que +\begin_inset Formula $x_{k_{1}}\in B(a_{i_{0}},\epsilon)$ +\end_inset + +, entonces +\begin_inset Formula $\Vert x_{k_{1}+1}-a_{i_{0}}\Vert\leq\Vert x_{k_{1}+1}-x_{k_{1}}\Vert+\Vert x_{k_{1}}-a_{i_{0}}\Vert<2\epsilon$ +\end_inset + +, y por la desigualdad triangular, +\begin_inset Formula $\Vert x_{k_{1}+1}-a_{i}\Vert>\varepsilon$ +\end_inset + + para +\begin_inset Formula $i\neq i_{0}$ +\end_inset + +. + Por tanto +\begin_inset Formula $x_{k_{1}+1}\in B(a_{i_{0}},\varepsilon)$ +\end_inset + +, y por inducción, +\begin_inset Formula $x_{k}\in B(a_{i_{0}},\varepsilon)$ +\end_inset + + para todo +\begin_inset Formula $k\geq k_{1}$ +\end_inset + +, con lo que solo hay un punto de acumulación, +\begin_inset Formula $a_{i_{0}}$ +\end_inset + +, y entonces +\begin_inset Formula $\lim_{k}x_{k}=a_{i_{0}}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Para la primera parte del teorema, sean +\begin_inset Formula $A_{k}=:(a_{kij})_{ij}$ +\end_inset + + y +\begin_inset Formula $\varepsilon_{k}:=\sum_{i\neq j}(a_{kij})^{2}$ +\end_inset + +. + Dados los +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $q$ +\end_inset + + de la iteración +\begin_inset Formula $k$ +\end_inset + +, restringiéndonos a la submatriz de +\begin_inset Formula $A$ +\end_inset + + formada por las filas y columnas +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $q$ +\end_inset + +, la suma de los elementos de los 4 coeficientes se conserva tras el giro, + luego +\begin_inset Formula $2a_{kpq}^{2}+a_{kpp}^{2}+a_{kqq}^{2}=a_{(k+1)pp}^{2}+a_{(k+1)qq}^{2}$ +\end_inset + +. + Además, la suma de los cuadrados de los elementos de +\begin_inset Formula $A$ +\end_inset + + no cambia, y tampoco cambian los elementos de su diagonal distintos de + +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $q$ +\end_inset + +, y como +\begin_inset Formula $\varepsilon_{k}$ +\end_inset + + es la suma de los cuadrados de los elementos de +\begin_inset Formula $A_{k}$ +\end_inset + + fuera de la diagonal, +\begin_inset Formula $\varepsilon_{k+1}=\varepsilon_{k}-2(a_{kpq})^{2}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Como +\begin_inset Formula $a_{pq}=\max_{i\neq j}|a_{ij}|$ +\end_inset + +, +\begin_inset Formula $\varepsilon_{k}\leq n(n-1)a_{kpq}^{2}$ +\end_inset + +, pues +\begin_inset Formula $n(n-1)$ +\end_inset + + es el número de elementos fuera de la diagonal principal. + Así, +\begin_inset Formula $a_{kpq}^{2}\geq\frac{\varepsilon_{k}}{n(n-1)}$ +\end_inset + + y +\begin_inset Formula +\[ +\varepsilon_{k+1}\leq\left(1-\frac{2}{n(n-1)}\right)\varepsilon_{k}, +\] + +\end_inset + +de donde +\begin_inset Formula $\lim_{k}\varepsilon_{k}=0$ +\end_inset + + y los elementos de +\begin_inset Formula $A_{k}$ +\end_inset + + fuera de la diagonal convergen a 0, y queda ver que los elementos de la + diagonal también convergen. + +\end_layout + +\begin_layout Standard +Sea +\begin_inset Formula $D_{k}:=\text{diag}(a_{k11},\dots,a_{knn})$ +\end_inset + +. + Si +\begin_inset Formula $(D_{k_{m}})_{m}$ +\end_inset + + una subsucesión de +\begin_inset Formula $(D_{k})_{k}$ +\end_inset + +, por continuidad, y como los elementos de +\begin_inset Formula $A_{k_{m}}$ +\end_inset + + fuera de la diagonal convergen a 0, +\begin_inset Formula $\lim_{m}\det(\lambda I_{n}-A_{k_{m}})=\det(\lambda I_{n}-D)$ +\end_inset + +. + Pero +\begin_inset Formula $A_{k}$ +\end_inset + + y +\begin_inset Formula $A$ +\end_inset + + son semejantes, luego tienen el mismo polinomio característico y, por tanto, + este coincide con el de +\begin_inset Formula $D$ +\end_inset + +. + Así, los elementos de la diagonal de +\begin_inset Formula $D$ +\end_inset + + son los valores propios de +\begin_inset Formula $A$ +\end_inset + + y con las mismas multiplicidades. + Por tanto, los puntos de acumulación de +\begin_inset Formula $(D_{k})_{k}$ +\end_inset + + son las diagonales formadas por los valores propios de +\begin_inset Formula $A$ +\end_inset + + en distinto orden, de las que hay un máximo de +\begin_inset Formula $n!$ +\end_inset + +, y en particular +\begin_inset Formula $(D_{k})_{k}$ +\end_inset + + tiene una cantidad finita de puntos de acumulación. +\end_layout + +\begin_layout Standard +Tenemos que +\begin_inset Formula $\lim_{k}(D_{k+1}-D_{k})=0$ +\end_inset + +. + En efecto, +\begin_inset Formula +\[ +a_{(k+1)ii}-a_{kii}=\begin{cases} +0, & i\neq p,q;\\ +-\tan\theta_{k}a_{kpq}, & i=p;\\ +\tan\theta_{k}a_{kpq}, & i=q; +\end{cases} +\] + +\end_inset + +pero +\begin_inset Formula $|\tan\theta_{k}|\leq1$ +\end_inset + + por ser +\begin_inset Formula $\theta_{k}\in(-\frac{\pi}{4},\frac{\pi}{4}]$ +\end_inset + +, y +\begin_inset Formula $|a_{kpq}|\leq\sqrt{\varepsilon_{k}}\to0$ +\end_inset + +. + Además, +\begin_inset Formula $(D_{k})_{k}$ +\end_inset + + está acotada, pues +\begin_inset Formula $\Vert D_{k}\Vert_{E}\leq\Vert A_{k}\Vert_{E}=\Vert A\Vert_{E}$ +\end_inset + +. + Aplicando la propiedad al principio, +\begin_inset Formula $(D_{k})_{k}$ +\end_inset + + converge a una diagonal formada por los valores propios de +\begin_inset Formula $A$ +\end_inset + + en algún orden, y por tanto +\begin_inset Formula $(A_{k})_{k}$ +\end_inset + + también. +\end_layout + +\begin_layout Standard +Para la segunda parte, sea +\begin_inset Formula $(\Omega_{k_{m}})_{m}$ +\end_inset + + una subsucesión de +\begin_inset Formula $(\Omega_{k})_{k}$ +\end_inset + + que converge a un cierto punto acumulación +\begin_inset Formula $P$ +\end_inset + + de +\begin_inset Formula $(\Omega_{k})_{k}$ +\end_inset + +, entonces +\begin_inset Formula $\Omega_{k_{m}}^{t}\to P^{t}$ +\end_inset + + e +\begin_inset Formula $I_{n}=\Omega_{k_{m}}^{t}\Omega_{k_{m}}\to P^{t}P$ +\end_inset + +, luego +\begin_inset Formula $P^{t}P=I_{n}$ +\end_inset + + y +\begin_inset Formula $P$ +\end_inset + + es ortogonal. + Como +\begin_inset Formula $\Omega_{k}^{t}A\Omega_{k}\to D$ +\end_inset + +, +\begin_inset Formula $P^{t}AP=D$ +\end_inset + +, lo que implica que las columnas de +\begin_inset Formula $P$ +\end_inset + + forman una base ortonormal de vectores propios asociados a los valores + propios en +\begin_inset Formula $D$ +\end_inset + +. + Como estamos suponiendo que todos los valores propios son distintos, cada + uno tiene un subespacio propio de dimensión 1 y hay exactamente dos vectores + propios ortonormales, uno opuesto del otro, para cada valor propio, pudiendo + escribir +\begin_inset Formula +\[ +P=:\begin{pmatrix}| & & |\\ +\pm p_{1} & \cdots & \pm p_{n}\\ +| & & | +\end{pmatrix}, +\] + +\end_inset + +con lo que los puntos de acumulación de +\begin_inset Formula $(\Omega_{k})_{k}$ +\end_inset + + solo se diferencian en el signo de las columnas y por tanto hay un máximo + de +\begin_inset Formula $2^{n}$ +\end_inset + +, en particular una cantidad finita. +\end_layout + +\begin_layout Standard +Dada una matriz ortogonal +\begin_inset Formula $O$ +\end_inset + +, +\begin_inset Formula $\Vert O\Vert_{2}=1$ +\end_inset + +, y como todas las normas en +\begin_inset Formula ${\cal M}_{n}(\mathbb{R})\cong\mathbb{R}^{n^{2}}$ +\end_inset + + son equivalentes, existe +\begin_inset Formula $\beta>0$ +\end_inset + + tal que +\begin_inset Formula $\Vert O\Vert_{E}\leq\beta\Vert O\Vert_{2}=\beta$ +\end_inset + +. + Por tanto +\begin_inset Formula $(\Omega_{k})_{k}$ +\end_inset + + está acotada. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $\theta_{k}$ +\end_inset + + es tal que +\begin_inset Formula $t=\tan(2\theta_{k})$ +\end_inset + + en la iteración +\begin_inset Formula $k$ +\end_inset + +, en esta iteración, +\begin_inset Formula +\[ +\tan(2\theta_{k})=\frac{2a_{kpq}}{a_{kqq}-a_{kpp}}. +\] + +\end_inset + +Como cada +\begin_inset Formula $(a_{kii})_{k}$ +\end_inset + + converge a un valor propio, existe +\begin_inset Formula $k_{0}$ +\end_inset + + tal que, para +\begin_inset Formula $k\geq k_{0}$ +\end_inset + +, +\begin_inset Formula +\[ +\min_{i\neq j}|a_{kii}-a_{kjj}|\geq\frac{1}{2}\min_{i\neq j}|\lambda_{i}-\lambda_{j}|=:M>0, +\] + +\end_inset + +con lo que +\begin_inset Formula $|a_{kqq}-a_{kpp}|\geq M$ +\end_inset + + y, como todos los elementos de +\begin_inset Formula $A_{k}$ +\end_inset + + fuera de la diagonal principal tienden a 0, +\begin_inset Formula $(a_{kpq})_{k}$ +\end_inset + + tiende a cero (aunque +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $q$ +\end_inset + + cambien según +\begin_inset Formula $k$ +\end_inset + +), +\begin_inset Formula $\tan\theta_{k}\to0$ +\end_inset + + y, como +\begin_inset Formula $|\theta_{k}|\leq\frac{\pi}{4}$ +\end_inset + +, +\begin_inset Formula $\theta_{k}\to0$ +\end_inset + +, luego si +\begin_inset Formula $O_{k}$ +\end_inset + + es el giro tal que +\begin_inset Formula $\Omega_{k+1}=\Omega_{k}O_{k}$ +\end_inset + +, +\begin_inset Formula $O_{k}\to I_{n}$ +\end_inset + + y, por tanto, +\begin_inset Formula $\lim_{k}(\Omega_{k+1}-\Omega_{k})=\lim_{k}(O_{k}-I_{n})\Omega_{k}=0$ +\end_inset + +, pues +\begin_inset Formula $(\Omega_{k})_{k}$ +\end_inset + + está acotada. + Con esto, y aplicando la propiedad del principio, +\begin_inset Formula $(\Omega_{k})_{k}$ +\end_inset + + converge a una matriz, cuyas columnas formaran una base ortonormal de vectores + propios en el mismo orden que los valores propios de +\begin_inset Formula $D$ +\end_inset + +. +\end_layout + +\begin_layout Section +Método QR +\end_layout + +\begin_layout Standard +Dada una matriz +\begin_inset Formula $A\in{\cal M}_{n}$ +\end_inset + +, definimos la sucesión +\begin_inset Formula $(A_{k})_{k}$ +\end_inset + + como +\begin_inset Formula $A_{0}:=A$ +\end_inset + + y +\begin_inset Formula $A_{k+1}:=R_{k}Q_{k}$ +\end_inset + +, donde +\begin_inset Formula $(Q_{k},R_{k})$ +\end_inset + + es la descomposición QR de +\begin_inset Formula $A_{k}$ +\end_inset + +. + Bajo ciertas condiciones, esta sucesión tiende a una matriz triangular + superior, con los valores propios en la diagonal. +\end_layout + +\begin_layout Standard +Para obtener una aproximación de los valores propios a partir de una aproximació +n +\begin_inset Formula $A_{p}:=(u_{ij})$ +\end_inset + + de dicha matriz, definimos una matriz +\begin_inset Formula $V:=(v_{ij})\in{\cal M}_{n}$ +\end_inset + + dada por +\begin_inset Formula +\[ +v_{ij}:=\begin{cases} +1, & i=j;\\ +0, & i>j;\\ +{\displaystyle -\frac{1}{u_{ii}-u_{jj}}\sum_{k=i+1}^{j}u_{ik}v_{kj}}, & i<j; +\end{cases} +\] + +\end_inset + +y los vectores propios son las columnas de +\begin_inset Formula $Q_{1}\cdots Q_{p}V$ +\end_inset + +. +\end_layout + +\end_body +\end_document |