#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
\use_default_options true
\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 Section
Matrices
\end_layout

\begin_layout Standard
\begin_inset Note Comment
status open

\begin_layout Plain Layout
Una 
\series bold
matriz
\series default
 de tamaño 
\begin_inset Formula $m\times n$
\end_inset

, o de 
\begin_inset Formula $m$
\end_inset

 filas y 
\begin_inset Formula $n$
\end_inset

 columnas, sobre un anillo 
\begin_inset Formula $A$
\end_inset

 es una disposición de elementos de 
\begin_inset Formula $A$
\end_inset

 ordenados por un índice de 1 a 
\begin_inset Formula $m$
\end_inset

 y otro de 1 a 
\begin_inset Formula $n$
\end_inset

, que se representa como
\begin_inset Formula 
\[
\left(\begin{array}{ccc}
a_{11} & \cdots & a_{1n}\\
\vdots &  & \vdots\\
a_{m1} & \cdots & a_{mn}
\end{array}\right).
\]

\end_inset

Llamamos 
\begin_inset Formula ${\cal M}_{m\times n}(A)$
\end_inset

 o 
\begin_inset Formula ${\cal M}_{mn}(A)$
\end_inset

 al conjunto de las matrices 
\begin_inset Formula $m\times n$
\end_inset

 sobre el anillo 
\begin_inset Formula $A$
\end_inset

, o 
\begin_inset Formula ${\cal M}_{n}(A)\coloneqq {\cal M}_{nn}(A)$
\end_inset

, pudiendo omitir 
\begin_inset Formula $(A)$
\end_inset

 si el anillo está claro.
 Una matriz es 
\series bold
fila
\series default
 si solo tiene una fila o 
\series bold
columna
\series default
 si solo tiene una columna, y podemos identificar 
\begin_inset Formula ${\cal M}_{1n}(A)$
\end_inset

 o 
\begin_inset Formula ${\cal M}_{n1}(A)$
\end_inset

 con 
\begin_inset Formula $A^{n}$
\end_inset

.
\end_layout

\begin_layout Plain Layout
Dadas 
\begin_inset Formula $X,Y\in{\cal M}_{m\times n}(A)$
\end_inset

, llamamos 
\begin_inset Formula $X+Y\coloneqq (X_{ij}+Y_{ij})_{1\leq i\leq m}^{1\leq j\leq n}$
\end_inset

, y dadas 
\begin_inset Formula $X\in{\cal M}_{m\times n}(A)$
\end_inset

 e 
\begin_inset Formula $Y\in{\cal M}_{n\times p}(A)$
\end_inset

, llamamos 
\begin_inset Formula $XY\coloneqq (\sum_{k=1}^{n}X_{ik}Y_{kj})_{1\leq i\leq m}^{1\leq j\leq p}$
\end_inset

.
 Con esto, 
\begin_inset Formula ${\cal M}_{n}(A)$
\end_inset

 es un anillo con elemento nulo 
\begin_inset Formula $(0)_{ij}$
\end_inset

 y elemento unidad la 
\series bold
matriz identidad
\series default
, 
\begin_inset Formula $(\delta_{ij})_{ij}$
\end_inset

.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Dada 
\begin_inset Formula $M\in{\cal M}_{m\times n}(\mathbb{C})$
\end_inset

, llamamos 
\series bold
matriz adjunta
\series default
 de 
\begin_inset Formula $M$
\end_inset

 a 
\begin_inset Formula $M^{*}\coloneqq (\overline{M_{ji}})_{ij}\in{\cal M}_{n\times m}(\mathbb{C})$
\end_inset

 y 
\series bold
matriz traspuesta
\series default
 de 
\begin_inset Formula $M$
\end_inset

 a 
\begin_inset Formula $M^{t}\coloneqq (M_{ji})_{ij}\in{\cal M}_{n\times m}(\mathbb{C})$
\end_inset

, que coincide con la adjunta cuando los coeficientes son reales, y se tiene
 
\begin_inset Formula $(AB)^{*}=B^{*}A^{*}$
\end_inset

 y 
\begin_inset Formula $A^{**}=A$
\end_inset

.
 
\end_layout

\begin_layout Standard
Llamamos 
\series bold
traza
\series default
 de una matriz 
\begin_inset Formula $A\in{\cal M}_{n}$
\end_inset

 a
\begin_inset Formula 
\[
\text{tr}A:=\sum_{k=1}^{n}A_{kk}.
\]

\end_inset

El 
\series bold
determinante
\series default
 es la única aplicación 
\begin_inset Formula $\det:{\cal M}_{n}(\mathbb{K})\to\mathbb{K}$
\end_inset

 multilineal (lineal en cada fila o columna) y alternada (que cambia de
 signo al permutar dos filas o columnas) que le asocia 1 a la identidad,
 y cumple 
\begin_inset Formula $\det(AB)=\det(A)\det(B)$
\end_inset

.
\end_layout

\begin_layout Standard
Sean 
\begin_inset Formula $A\in{\cal M}_{n}(\mathbb{K})$
\end_inset

, 
\begin_inset Formula $\lambda\in\mathbb{K}$
\end_inset

 y 
\begin_inset Formula $p\in\mathbb{K}^{n}\setminus0$
\end_inset

, si 
\begin_inset Formula $Ap=\lambda p$
\end_inset

, 
\begin_inset Formula $\lambda$
\end_inset

 es un 
\series bold
valor propio
\series default
 y 
\begin_inset Formula $p$
\end_inset

 es un 
\series bold
vector propio
\series default
 de 
\begin_inset Formula $A$
\end_inset

.
 Los valores propios de 
\begin_inset Formula $A$
\end_inset

 son los ceros de su 
\series bold
polinomio característico
\series default
, 
\begin_inset Formula $p_{A}(\lambda)\coloneqq \det(A-\lambda I)$
\end_inset

.
 Si estos son 
\begin_inset Formula $\lambda_{1},\dots,\lambda_{n}$
\end_inset

, el 
\series bold
espectro
\series default
 de 
\begin_inset Formula $A$
\end_inset

 es 
\begin_inset Formula $\sigma(A)\coloneqq \{\lambda_{1},\dots,\lambda_{n}\}$
\end_inset

 y su 
\series bold
radio espectral
\series default
 es 
\begin_inset Formula $\rho(A)\coloneqq \max\{|\lambda_{1}|,\dots,|\lambda_{n}|\}$
\end_inset

.
\end_layout

\begin_layout Section
Sistemas de ecuaciones
\end_layout

\begin_layout Standard
Un 
\series bold
sistema
\series default
 de 
\begin_inset Formula $m$
\end_inset

 ecuaciones lineales con 
\begin_inset Formula $n$
\end_inset

 incógnitas es uno de la forma
\begin_inset Formula 
\[
\left\{ \begin{aligned}a_{11}x_{1}+\dots+a_{1n}x_{n} & =b_{1},\\
 & \vdots\\
a_{m1}x_{1}+\dots+a_{mn}x_{n} & =b_{m}.
\end{aligned}
\right.
\]

\end_inset

Llamamos 
\series bold
coeficientes
\series default
 a los escalares 
\begin_inset Formula $a_{ij}$
\end_inset

, 
\series bold
términos independientes
\series default
 a los escalares 
\begin_inset Formula $b_{i}$
\end_inset

 y 
\series bold
soluciones
\series default
 del sistema a los vectores 
\begin_inset Formula $(x_{1},\dots,x_{n})$
\end_inset

 que cumplen todas las igualdades.
 Un sistema es 
\series bold
compatible
\series default
 si tiene soluciones, en cuyo caso es 
\series bold
determinado
\series default
 si solo tiene una o 
\series bold
indeterminado
\series default
 si tiene más, y en otro caso es 
\series bold
incompatible
\series default
.
 Es 
\series bold
homogéneo
\series default
 si todos los términos independientes son 0, en cuyo caso tiene al menos
 la 
\series bold
solución trivial
\series default
 0.
 Dos sistemas de ecuaciones son 
\series bold
equivalentes
\series default
 si tienen el mismo conjunto de soluciones.
\end_layout

\begin_layout Standard
Llamamos 
\series bold
matriz de coeficientes
\series default
 a la matriz 
\begin_inset Formula $m\times n$
\end_inset

 
\begin_inset Formula $A\coloneqq (a_{ij})_{ij}$
\end_inset

, 
\series bold
columna de términos independientes
\series default
 a la matriz columna 
\begin_inset Formula $b\coloneqq (b_{i})_{ij}$
\end_inset

 y 
\series bold
matriz ampliada
\series default
 a 
\begin_inset Formula $\left(\begin{array}{c|c}
A & b\end{array}\right)$
\end_inset

.
 Entonces podemos representar el sistema como 
\begin_inset Formula $Ax=b$
\end_inset

.
 Si 
\begin_inset Formula $A$
\end_inset

 es invertible, el sistema es compatible determinado, pues 
\begin_inset Formula $x=A^{-1}Ax=A^{-1}b$
\end_inset

.
\end_layout

\begin_layout Section
Aplicaciones lineales
\end_layout

\begin_layout Standard
Una 
\series bold
base
\series default
 de un 
\begin_inset Formula $\mathbb{K}$
\end_inset

-espacio vectorial 
\begin_inset Formula $E$
\end_inset

 de dimensión finita es una tupla 
\begin_inset Formula $(v_{1},\dots,v_{n})$
\end_inset

 de vectores linealmente independientes de 
\begin_inset Formula $E$
\end_inset

 tal que todo 
\begin_inset Formula $x\in E$
\end_inset

 se puede escribir como
\begin_inset Formula 
\[
\sum_{k=1}^{n}x_{k}v_{k}
\]

\end_inset

con 
\begin_inset Formula $x_{1},\dots,x_{n}\in\mathbb{K}$
\end_inset

.
 Esto nos permite identificar los vectores 
\begin_inset Formula $x\in E$
\end_inset

 con sus coordenadas 
\begin_inset Formula $(x_{1},\dots,x_{n})\in\mathbb{K}^{n}$
\end_inset

, y con la correspondiente matriz columna.
\end_layout

\begin_layout Standard
Un 
\series bold
producto escalar
\series default
 en un 
\begin_inset Formula $\mathbb{R}$
\end_inset

-espacio vectorial 
\begin_inset Formula $E$
\end_inset

 es una función 
\begin_inset Formula $\langle\cdot,\cdot\rangle:E\times E\to\mathbb{R}$
\end_inset

 bilineal simétrica tal que 
\begin_inset Formula $\forall f\in E\setminus0,\langle f,f\rangle>0$
\end_inset

.
 
\end_layout

\begin_layout Standard
Llamamos 
\series bold
producto escalar euclídeo
\series default
 en 
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset

 a
\begin_inset Formula 
\[
\langle x,y\rangle:=\sum_{k=1}^{n}x_{k}y_{k}=x^{t}y=y^{t}x,
\]

\end_inset


\series bold
producto escalar hermitiano
\series default
 en 
\begin_inset Formula $\mathbb{C}^{n}$
\end_inset

 a
\begin_inset Formula 
\[
\langle x,y\rangle:=\sum_{k=1}^{n}x_{k}\overline{y_{k}}=y^{*}x=\overline{x^{*}y}.
\]

\end_inset


\end_layout

\begin_layout Standard
Sean 
\begin_inset Formula $M\in{\cal M}_{m\times n}(\mathbb{C})$
\end_inset

, 
\begin_inset Formula $u\in\mathbb{C}^{n}$
\end_inset

 y 
\begin_inset Formula $v\in\mathbb{C}^{m}$
\end_inset

, 
\begin_inset Formula $\langle Mu,v\rangle=\langle u,M^{*}v\rangle$
\end_inset

, y en particular, para 
\begin_inset Formula $M\in{\cal M}_{m\times n}(\mathbb{R})$
\end_inset

, 
\begin_inset Formula $u\in\mathbb{R}^{n}$
\end_inset

 y 
\begin_inset Formula $v\in\mathbb{R}^{m}$
\end_inset

, 
\begin_inset Formula $\langle Mu,v\rangle=\langle u,M^{t}v\rangle$
\end_inset

.
 En efecto, 
\begin_inset Formula $\langle Me_{i},e_{j}\rangle=\langle(M_{ki})_{k},e_{j}\rangle=M_{ji}$
\end_inset

, y 
\begin_inset Formula $\langle e_{i},M^{*}e_{j}\rangle=\langle e_{i},((M^{*})_{kj})_{k}\rangle=\langle e_{i},(\overline{M_{jk}})_{k}\rangle=\overline{\overline{M_{ji}}}=M_{ji}$
\end_inset

.
\end_layout

\begin_layout Standard
Dos vectores 
\begin_inset Formula $x,y\in\mathbb{C}^{n}$
\end_inset

 son 
\series bold
ortogonales
\series default
 si 
\begin_inset Formula $\langle x,y\rangle=0$
\end_inset

.
\end_layout

\begin_layout Section
Matrices especiales
\end_layout

\begin_layout Standard
Una matriz 
\begin_inset Formula $A$
\end_inset

 es 
\series bold
cuadrada
\series default
 si tiene el mismo número de filas que de columnas, en cuyo caso es 
\series bold
diagonal
\series default
 si 
\begin_inset Formula $\forall i\neq j,A_{ij}=0$
\end_inset

, 
\series bold
triangular superior
\series default
 si 
\begin_inset Formula $\forall i>j,A_{ij}=0$
\end_inset

 y 
\series bold
triangular inferior
\series default
 si 
\begin_inset Formula $\forall i<j,A_{ij}=0$
\end_inset

.
 Es 
\series bold
simétrica
\series default
 si 
\begin_inset Formula $A=A^{t}$
\end_inset

, 
\series bold
hermitiana
\series default
 si 
\begin_inset Formula $A=A^{*}$
\end_inset

, 
\series bold
ortogonal
\series default
 si 
\begin_inset Formula $A^{-1}=A^{t}$
\end_inset

, 
\series bold
unitaria
\series default
 si 
\begin_inset Formula $A^{-1}=A^{*}$
\end_inset

 y 
\series bold
normal
\series default
 si 
\begin_inset Formula $AA^{*}=A^{*}A$
\end_inset

.
 Si 
\begin_inset Formula $A\in{\cal M}_{n}(\mathbb{C})$
\end_inset

:
\end_layout

\begin_layout Enumerate
\begin_inset CommandInset label
LatexCommand label
name "enu:unitary"

\end_inset

Existe 
\begin_inset Formula $U\in{\cal M}_{n}$
\end_inset

 unitaria tal que 
\begin_inset Formula $U^{-1}AU$
\end_inset

 es triangular superior.
\end_layout

\begin_deeper
\begin_layout Standard
Lo probamos primero para 
\begin_inset Formula $U$
\end_inset

 cualquiera.
 Para 
\begin_inset Formula $n=1$
\end_inset

 esto es claro.
 Sea ahora 
\begin_inset Formula $n>1$
\end_inset

 y supongamos esto probado para 
\begin_inset Formula $n-1$
\end_inset

.
 Si 
\begin_inset Formula $f:\mathbb{C}^{n}\to\mathbb{C}^{n}$
\end_inset

 es la aplicación lineal asociada a 
\begin_inset Formula $f$
\end_inset

, por el teorema fundamental del álgebra, el polinomio característico de
 
\begin_inset Formula $A$
\end_inset

 tendrá raíz y por tanto 
\begin_inset Formula $f$
\end_inset

 tendrá un valor propio 
\begin_inset Formula $\lambda\in\mathbb{C}$
\end_inset

 con vector propio asociado 
\begin_inset Formula $v_{1}$
\end_inset

.
 Sean 
\begin_inset Formula $p_{2},\dots,p_{n}$
\end_inset

 tales que 
\begin_inset Formula $(v_{1},p_{2},\dots,p_{n})$
\end_inset

 es base de 
\begin_inset Formula $\mathbb{C}^{n}$
\end_inset

 y 
\begin_inset Formula $W\coloneqq \text{span}(p_{2},\dots,p_{n})$
\end_inset

, existen 
\begin_inset Formula $g:W\to W$
\end_inset

 y 
\begin_inset Formula $\alpha_{2},\dots,\alpha_{n}\in\mathbb{C}$
\end_inset

 tales que, para 
\begin_inset Formula $2\leq k\leq n$
\end_inset

, 
\begin_inset Formula $f(p_{k})=\alpha_{k}v_{1}+g(p_{k})$
\end_inset

.
 
\end_layout

\begin_layout Standard
Por la hipótesis de inducción, existe una base 
\begin_inset Formula $(v_{2},\dots,v_{n})$
\end_inset

 de 
\begin_inset Formula $W$
\end_inset

 en la que la matriz de 
\begin_inset Formula $g$
\end_inset

 es triangular superior.
 Si, para 
\begin_inset Formula $2\leq i\leq n$
\end_inset

, 
\begin_inset Formula $v_{i}=:\sum_{j=2}^{n}\gamma_{ij}p_{j}$
\end_inset

, como 
\begin_inset Formula $f(v_{1})=\lambda v_{1}$
\end_inset

 y, para 
\begin_inset Formula $2\leq i\leq n$
\end_inset

, 
\begin_inset Formula $f(v_{i})=\left(\sum_{k=2}^{n}\alpha_{k}\gamma_{ik}\right)v_{1}+g(v_{i})$
\end_inset

, tenemos que la matriz 
\begin_inset Formula $(b_{ij})$
\end_inset

 de 
\begin_inset Formula $f$
\end_inset

 con la base 
\begin_inset Formula $(v_{1},\dots,v_{n})$
\end_inset

 es triangular superior.
 
\end_layout

\begin_layout Standard
Por el método de Gram-Schmidt, existe una base ortonormal 
\begin_inset Formula $(u_{1},\dots,u_{n})$
\end_inset

 tal que, para 
\begin_inset Formula $1\leq k\leq n$
\end_inset

, 
\begin_inset Formula $\text{span}(u_{1},\dots,u_{k})=\text{span}(v_{1},\dots,v_{k})$
\end_inset

, y como 
\begin_inset Formula $f(v_{k})=\sum_{i=1}^{j}b_{ik}v_{i}$
\end_inset

 es combinación lineal de 
\begin_inset Formula $(v_{1},\dots,v_{k})$
\end_inset

, también lo es de 
\begin_inset Formula $(u_{1},\dots,u_{k})$
\end_inset

 y 
\begin_inset Formula $f$
\end_inset

 se expresa en la base 
\begin_inset Formula $(u_{1},\dots,u_{n})$
\end_inset

 como matriz triangular.
 Como esta base es ortonormal respecto al producto escalar hermitiano, la
 matriz de paso es unitaria.
\end_layout

\end_deeper
\begin_layout Enumerate
Si 
\begin_inset Formula $A$
\end_inset

 es normal, existe 
\begin_inset Formula $U\in{\cal M}_{n}$
\end_inset

 unitaria tal que 
\begin_inset Formula $U^{-1}AU$
\end_inset

 es diagonal.
\end_layout

\begin_deeper
\begin_layout Standard
Existe 
\begin_inset Formula $U$
\end_inset

 unitaria tal que 
\begin_inset Formula $T\coloneqq U^{-1}AU=U^{*}AU$
\end_inset

 es triangular superior, pero 
\begin_inset Formula $T^{*}T=(U^{*}AU)^{*}U^{*}AU=U^{*}A^{*}U^{**}U^{*}AU=U^{*}A^{*}UU^{*}AU=U^{*}A^{*}AU$
\end_inset

 y 
\begin_inset Formula $TT^{*}=U^{*}AU(U^{*}AU)^{*}=U^{*}AUU^{*}A^{*}U=U^{*}AA^{*}U=U^{*}A^{*}AU=T^{*}T$
\end_inset

, luego 
\begin_inset Formula $T$
\end_inset

 es normal.
 Entonces, para 
\begin_inset Formula $i<j$
\end_inset

, 
\begin_inset Formula $A_{ij}=(A^{*})_{ij}=\overline{A_{ji}}=\overline{0}=0$
\end_inset

, pues 
\begin_inset Formula $j>i$
\end_inset

 y 
\begin_inset Formula $A$
\end_inset

 es triangular superior.
 Por tanto 
\begin_inset Formula $T$
\end_inset

 es diagonal.
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sremember{AAlG}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Toda matriz simétrica real 
\begin_inset Formula $A\in{\cal M}_{m}(\mathbb{R})$
\end_inset

 admite una matriz ortogonal 
\begin_inset Formula $P$
\end_inset

 tal que 
\begin_inset Formula $P^{-1}AP=P^{t}AP$
\end_inset

 es diagonal.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
eremember
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Dada 
\begin_inset Formula $A\in{\cal M}_{n}$
\end_inset

, los valores propios de 
\begin_inset Formula $A^{*}A$
\end_inset

 son no negativos, pues si 
\begin_inset Formula $p\neq0$
\end_inset

, 
\begin_inset Formula $A^{*}Ap=\lambda p\implies\lambda\Vert p\Vert^{2}=\lambda\langle p,p\rangle=\lambda p^{*}p=p^{*}\lambda p=p^{*}A^{*}Ap=(Ap)^{*}(Ap)=\Vert Ap\Vert^{2}\geq0\implies\lambda\geq0$
\end_inset

.
 Llamamos 
\series bold
valores singulares
\series default
 de 
\begin_inset Formula $A$
\end_inset

 a las raíces cuadradas de estos valores propios.
 Entonces:
\end_layout

\begin_layout Enumerate
Para 
\begin_inset Formula $A\in\mathbb{C}^{n}$
\end_inset

 con valores singulares 
\begin_inset Formula $\mu_{1},\dots,\mu_{n}$
\end_inset

, existen 
\begin_inset Formula $U$
\end_inset

 y 
\begin_inset Formula $V$
\end_inset

 unitarias tales que 
\begin_inset Formula $U^{*}AV=\text{diag}(\mu_{1},\dots,\mu_{n})$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula $A^{*}A$
\end_inset

 es normal, pues 
\begin_inset Formula $(A^{*}A)^{*}=A^{*}A$
\end_inset

 y por tanto 
\begin_inset Formula $(A^{*}A)(A^{*}A)^{*}=(A^{*}A)^{*}(A^{*}A)$
\end_inset

.
 Por el teorema anterior, existe 
\begin_inset Formula $V$
\end_inset

 unitaria tal que 
\begin_inset Formula $V^{*}A^{*}AV=\text{diag}(\mu_{1}^{2},\dots,\mu_{n}^{2})$
\end_inset

, donde los 
\begin_inset Formula $\mu_{i}$
\end_inset

 son los valores singulares de 
\begin_inset Formula $A$
\end_inset

.
 Si 
\begin_inset Formula $f_{1},\dots,f_{n}$
\end_inset

 son las columnas de 
\begin_inset Formula $AV$
\end_inset

, entonces 
\begin_inset Formula $f_{i}^{*}f_{j}=\mu_{i}^{2}\delta_{ij}$
\end_inset

 para 
\begin_inset Formula $i,j\in\{1,\dots,n\}$
\end_inset

.
 Podemos suponer que los valores singulares nulos son 
\begin_inset Formula $\mu_{1},\dots,\mu_{r}$
\end_inset

, luego 
\begin_inset Formula $f_{1},\dots,f_{r}=0$
\end_inset

, y haciendo 
\begin_inset Formula $u_{j}\coloneqq \frac{f_{j}}{\mu_{j}}$
\end_inset

 para 
\begin_inset Formula $j\in\{r+1,\dots,n\}$
\end_inset

, queda 
\begin_inset Formula $u_{i}^{*}u_{j}=\delta_{ij}$
\end_inset

 para 
\begin_inset Formula $i,j\in\{r+1,\dots,n\}$
\end_inset

, es decir, 
\begin_inset Formula $\{u_{r+1},\dots,u_{n}\}$
\end_inset

 son ortogonales.
 Completando con vectores ortogonales 
\begin_inset Formula $u_{1},\dots,u_{r}$
\end_inset

 se obtiene una base ortonormal de 
\begin_inset Formula $\mathbb{C}^{n}$
\end_inset

, y llamando 
\begin_inset Formula $U$
\end_inset

 a la matriz con columnas 
\begin_inset Formula $(u_{1},\dots,u_{n})$
\end_inset

, queda que
\begin_inset Formula 
\[
(U^{*}AV)_{ij}=u_{i}^{*}f_{j}=\begin{cases}
0 & \text{si }1\leq j\leq r\\
\mu_{j}u_{i}^{*}u_{j} & \text{si }r+1\leq j\leq n
\end{cases}=\mu_{i}\delta_{ij}.
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Enumerate
Para 
\begin_inset Formula $A\in\mathbb{R}^{n}$
\end_inset

 con valores singulares 
\begin_inset Formula $\mu_{1},\dots,\mu_{n}$
\end_inset

, existen 
\begin_inset Formula $U$
\end_inset

 y 
\begin_inset Formula $V$
\end_inset

 ortogonales tales que 
\begin_inset Formula $U^{t}AV=\text{diag}(\mu_{1},\dots,\mu_{n})$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Análogo, viendo que 
\begin_inset Formula $A^{t}A$
\end_inset

 es simétrica y cambiando adjuntas por traspuestas y unitarias por ortogonales.
\end_layout

\end_deeper
\begin_layout Section
Cocientes de Rayleigh
\end_layout

\begin_layout Standard
El 
\series bold
cociente de Rayleigh
\series default
 de una matriz 
\begin_inset Formula $A\in{\cal M}_{n}(\mathbb{C})$
\end_inset

 es una aplicación 
\begin_inset Formula $R_{A}:\mathbb{C}^{n}\setminus0\to\mathbb{C}$
\end_inset

 dada por
\begin_inset Formula 
\[
R_{A}(v):=\frac{\langle Av,v\rangle}{\langle v,v\rangle}=\frac{v^{*}Av}{v^{*}v}.
\]

\end_inset

Si 
\begin_inset Formula $A$
\end_inset

 es hermitiana, este cociente toma valores reales, pues entonces 
\begin_inset Formula $\overline{R_{A}(v)}=\frac{\overline{\langle Av,v\rangle}}{\langle v,v\rangle}=\frac{\langle v,Av\rangle}{\langle v,v\rangle}=\frac{\langle A^{*}v,v\rangle}{\langle v,v\rangle}=\frac{\langle Av,v\rangle}{\langle v,v\rangle}=R_{A}(v)$
\end_inset

.
\end_layout

\begin_layout Standard
Sean 
\begin_inset Formula $A\in{\cal M}_{n}$
\end_inset

 hermitiana con valores propios 
\begin_inset Formula $\lambda_{1}\leq\dots\leq\lambda_{n}$
\end_inset

, 
\begin_inset Formula $(p_{1},\dots,p_{n})$
\end_inset

 una base ortonormal (
\begin_inset Formula $(p_{i},p_{j})=\delta_{ij}$
\end_inset

) de vectores propios correspondientes (
\begin_inset Formula $Ap_{k}=\lambda_{k}p_{k}$
\end_inset

), 
\begin_inset Formula $E_{k}\coloneqq \text{span}\{p_{1},\dots,p_{k}\}$
\end_inset

 para cada 
\begin_inset Formula $k$
\end_inset

 (
\begin_inset Formula $E_{0}=\{0\}$
\end_inset

) y 
\begin_inset Formula ${\cal S}_{k}$
\end_inset

 la familia de todos los subespacios de 
\begin_inset Formula $\mathbb{C}^{n}$
\end_inset

 con dimensión 
\begin_inset Formula $k$
\end_inset

.
 Entonces, para 
\begin_inset Formula $1\leq k\leq n$
\end_inset

:
\end_layout

\begin_layout Enumerate
\begin_inset Formula $\lambda_{k}=R_{A}(p_{k})$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Sean 
\begin_inset Formula $U$
\end_inset

 unitaria tal que 
\begin_inset Formula $D\coloneqq U^{*}AU=\text{diag}(\lambda_{1},\dots,\lambda_{n})$
\end_inset

, 
\begin_inset Formula $v\in\mathbb{C}^{n}\setminus0$
\end_inset

 y 
\begin_inset Formula $v=Uw$
\end_inset

, 
\begin_inset Formula 
\[
R_{A}(v)=\frac{v^{*}Av}{v^{*}v}=\frac{w^{*}U^{*}AUw}{w^{*}U^{*}Uw}=\frac{D}{w^{*}w}=\frac{\sum_{k}\lambda_{k}|w_{k}|^{2}}{\sum_{k}|w_{k}|^{2}}.
\]

\end_inset

 Como 
\begin_inset Formula $w$
\end_inset

 es 
\begin_inset Formula $v$
\end_inset

 expresado respecto de la base 
\begin_inset Formula $(p_{1},\dots,p_{n})$
\end_inset

, 
\begin_inset Formula $Up_{k}=e_{k}$
\end_inset

, luego 
\begin_inset Formula $R_{A}(p_{k})=\frac{\lambda_{k}}{1}=\lambda_{k}$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Enumerate
\begin_inset Formula $\lambda_{k}=\max_{v\in E_{k}\setminus0}R_{A}(v)$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Si 
\begin_inset Formula $v$
\end_inset

 es de la forma 
\begin_inset Formula $\sum_{i=1}^{k}\alpha_{i}p_{i}$
\end_inset

, 
\begin_inset Formula $w=(\alpha_{1},\dots,\alpha_{k},0,\dots,0)$
\end_inset

, y como los valores propios están ordenados, 
\begin_inset Formula 
\[
R_{A}(v)=\frac{\sum_{i=1}^{k}\lambda_{i}|\alpha_{i}|^{2}}{\sum_{i=1}^{k}|\alpha_{i}|^{2}}\leq\frac{\sum_{i=1}^{k}\lambda_{k}|\alpha_{k}|^{2}}{\sum_{i=1}^{k}|\alpha_{k}|^{2}}=\lambda_{k}=R_{A}(p_{k}).
\]

\end_inset


\end_layout

\end_deeper
\begin_layout Enumerate
\begin_inset Formula $\lambda_{k}=\min_{0\neq v\bot E_{k-1}}R_{A}(v)$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
En este caso, 
\begin_inset Formula $v$
\end_inset

 es de la forma 
\begin_inset Formula $\sum_{i=k}^{n}\alpha_{i}p_{i}$
\end_inset

, y el razonamiento es análogo al del punto anterior.
\end_layout

\end_deeper
\begin_layout Enumerate
\begin_inset Formula $\lambda_{k}=\min_{W\in{\cal S}_{k}}\max_{v\in W\setminus0}R_{A}(v)$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Enumerate
\begin_inset Argument item:1
status open

\begin_layout Plain Layout
\begin_inset Formula $\geq]$
\end_inset


\end_layout

\end_inset


\begin_inset Formula $\lambda_{k}=\max_{v\in E_{k}\setminus\{0\}}R_{A}(v)\overset{E_{k}\in{\cal S}_{k}}{\leq}\inf_{W\in{\cal S}_{k}}\max_{v\in W\setminus\{0\}}R_{A}(v)$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Argument item:1
status open

\begin_layout Plain Layout
\begin_inset Formula $\leq]$
\end_inset


\end_layout

\end_inset

Queremos ver que 
\begin_inset Formula $\forall W\in{\cal S}_{k},\lambda_{k}\leq\max_{v\in W\setminus\{0\}}R_{A}(v)$
\end_inset

.
 Si 
\begin_inset Formula $E_{k-1}^{\bot}\coloneqq \{v\in V\mid v\bot E_{k-1}\}$
\end_inset

, basta ver que para todo subespacio 
\begin_inset Formula $W$
\end_inset

 de dimensión 
\begin_inset Formula $k$
\end_inset

, 
\begin_inset Formula $W\cap E_{k-1}^{\bot}\neq0$
\end_inset

, pues entonces, para 
\begin_inset Formula $v\in(W\cap E_{k-1}^{\bot})\setminus0$
\end_inset

, como 
\begin_inset Formula $0\neq v\bot E_{k-1}$
\end_inset

, 
\begin_inset Formula $\lambda_{k}\leq\min_{0\neq v\bot E_{k-1}}R_{A}(v)$
\end_inset

.
 Pero como 
\begin_inset Formula $E_{k-1}^{\bot}$
\end_inset

 tiene dimensión 
\begin_inset Formula $n-k+1$
\end_inset

, por Grassmann, 
\begin_inset Formula $\dim(W\cap E_{k-1}^{\bot})=\dim W+\dim E_{k-1}^{\bot}-\dim(W\oplus E_{k-1}^{\bot})\leq\dim W+\dim E_{k-1}^{\bot}-\dim\mathbb{C}^{n}=k+n-k+1-n=1$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Enumerate
\begin_inset Formula $\lambda_{k}=\max_{E\in{\cal S}_{k-1}}\min_{0\neq v\bot E}R_{A}(v)$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Análogo.
\end_layout

\end_deeper
\begin_layout Section
Normas matriciales
\end_layout

\begin_layout Standard
Sea 
\begin_inset Formula $E$
\end_inset

 un espacio vectorial sobre 
\begin_inset Formula $\mathbb{R}$
\end_inset

 o 
\begin_inset Formula $\mathbb{C}$
\end_inset

, una 
\series bold
norma
\series default
 sobre 
\begin_inset Formula $E$
\end_inset

 es una aplicación 
\begin_inset Formula $\Vert\cdot\Vert:E\to[0,+\infty)$
\end_inset

 tal que:
\end_layout

\begin_layout Enumerate
\begin_inset Formula $\Vert x\Vert=0\iff x=0$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Formula $\Vert x+y\Vert\leq\Vert x\Vert+\Vert y\Vert$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Formula $\Vert ax\Vert=|a|\Vert x\Vert$
\end_inset

.
\end_layout

\begin_layout Standard
Llamamos 
\series bold
espacio vectorial normado
\series default
 al par 
\begin_inset Formula $(E,\Vert\cdot\Vert)$
\end_inset

.
 La función 
\begin_inset Formula $d(x,y)=\Vert x-y\Vert$
\end_inset

 es una distancia en 
\begin_inset Formula $E$
\end_inset

.
 Todas las normas en un espacio de dimensión finita son equivalentes, es
 decir, definen la misma topología.
 
\end_layout

\begin_layout Standard
Si 
\begin_inset Formula $E$
\end_inset

 es un 
\begin_inset Formula $\mathbb{R}$
\end_inset

-espacio vectorial con un producto escalar 
\begin_inset Formula $\langle\cdot,\cdot\rangle$
\end_inset

, 
\begin_inset Formula $\Vert\cdot\Vert:E\to\mathbb{R}$
\end_inset

 dada por 
\begin_inset Formula $\Vert f\Vert\coloneqq \sqrt{\langle f,f\rangle}$
\end_inset

 define una norma en 
\begin_inset Formula $E$
\end_inset

.
 Llamamos 
\series bold
norma 
\begin_inset Formula $p$
\end_inset


\series default
 de 
\begin_inset Formula $x\in\mathbb{C}^{n}$
\end_inset

 a
\begin_inset Formula 
\[
\Vert x\Vert_{p}:=\sqrt[p]{\sum_{k=1}^{n}|x_{k}|^{p}},
\]

\end_inset


\series bold
norma euclídea
\series default
 a 
\begin_inset Formula $\Vert x\Vert_{2}=\sqrt{\langle x,x\rangle}$
\end_inset

 y 
\series bold
norma infinito
\series default
 a
\begin_inset Formula 
\[
\Vert x\Vert_{\infty}:=\max{_{k=1}^{n}}|x_{k}|.
\]

\end_inset


\end_layout

\begin_layout Standard
Una 
\series bold
norma matricial
\series default
 en 
\begin_inset Formula ${\cal M}_{n}(\mathbb{K})$
\end_inset

, donde 
\begin_inset Formula $\mathbb{K}$
\end_inset

 es 
\begin_inset Formula $\mathbb{R}$
\end_inset

 o 
\begin_inset Formula $\mathbb{C}$
\end_inset

, es una que cumple 
\begin_inset Formula $\forall A,B\in{\cal M}_{n}(\mathbb{K}),\Vert AB\Vert\leq\Vert A\Vert\Vert B\Vert$
\end_inset

.
 Dada una norma 
\begin_inset Formula $\Vert\cdot\Vert$
\end_inset

 en 
\begin_inset Formula $\mathbb{K}^{n}$
\end_inset

, llamamos 
\series bold
norma matricial sub
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
-
\end_layout

\end_inset

or
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
-
\end_layout

\end_inset

di
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
-
\end_layout

\end_inset

na
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
-
\end_layout

\end_inset

da
\series default
 a la norma 
\begin_inset Formula $\Vert\cdot\Vert$
\end_inset

 a la norma matricial en 
\begin_inset Formula ${\cal M}_{n}(\mathbb{K})$
\end_inset

 dada por
\begin_inset Formula 
\[
\Vert A\Vert:=\sup_{x\in\mathbb{K}^{n}\setminus\{0\}}\frac{\Vert Ax\Vert}{\Vert x\Vert}=\sup_{\Vert x\Vert\leq1}\frac{\Vert Ax\Vert}{\Vert x\Vert}=\sup_{\Vert x\Vert=1}\Vert Ax\Vert.
\]

\end_inset

Entonces, para 
\begin_inset Formula $A\in{\cal M}_{n}(\mathbb{K})$
\end_inset

 y 
\begin_inset Formula $x\in\mathbb{K}^{n}$
\end_inset

, 
\begin_inset Formula $\Vert Ax\Vert\leq\Vert A\Vert\Vert x\Vert$
\end_inset

.
\end_layout

\begin_layout Standard
Sea 
\begin_inset Formula $A\coloneqq (a_{ij})_{ij}\in{\cal M}_{n}(\mathbb{C})$
\end_inset

:
\end_layout

\begin_layout Enumerate
\begin_inset Formula $\Vert A\Vert_{1}=\max_{j}\sum_{i}|a_{ij}|$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula $\sup\{\Vert Ax\Vert\mid\Vert x\Vert=1\}=\sup\{\sum_{k}|Ax|_{k}\mid\sum_{k}|x_{k}|=1\}=\sup\{\sum_{k,i}|a_{ki}||x_{i}|\mid\sum_{i}|x_{i}|=1\}$
\end_inset

.
 Sea 
\begin_inset Formula $j$
\end_inset

 tal que 
\begin_inset Formula $\max_{i}\sum_{k}|a_{ki}|=\sum_{k}|a_{kj}|$
\end_inset

, para 
\begin_inset Formula $x$
\end_inset

 con 
\begin_inset Formula $\sum_{i}|x_{i}|$
\end_inset

, 
\begin_inset Formula 
\[
\sum_{k,i}|a_{ki}||x_{i}|=\sum_{i}\left(|x_{i}|\sum_{k}|a_{ki}|\right)\leq\left(\sum_{i}|x_{i}|\right)\left(\sum_{k}|a_{kj}|\right)=\sum_{k}|a_{kj}|,
\]

\end_inset

luego 
\begin_inset Formula $\sup\{\sum_{k,i}|a_{ki}||x_{i}|\mid\sum_{i}|x_{i}|=1\}=\max_{i}\sum_{k}|a_{ki}|$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Enumerate
\begin_inset Formula $\Vert A\Vert_{2}=\sqrt{\rho(A^{*}A)}=\Vert A^{*}\Vert_{2}$
\end_inset

.
 En particular, 
\begin_inset Formula $\Vert A\Vert_{2}$
\end_inset

 es el mayor valor singular de 
\begin_inset Formula $A$
\end_inset

, y si 
\begin_inset Formula $A$
\end_inset

 es unitaria o real ortogonal, 
\begin_inset Formula $\Vert A\Vert_{2}=1$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula 
\[
\Vert A\Vert_{2}^{2}=\sup\left\{ \frac{\Vert Ax\Vert_{2}^{2}}{\Vert x\Vert_{2}^{2}}\;\middle|\;\Vert x\Vert_{2}=1\right\} =\sup\left\{ \frac{\langle Ax,Ax\rangle}{\langle x,x\rangle}=\frac{\langle A^{*}Ax,x\rangle}{\langle x,x\rangle}=R_{A^{*}A}(x)\;\middle|\;\Vert x\Vert_{2}=1\right\} ,
\]

\end_inset

 pero si 
\begin_inset Formula $\lambda_{1},\dots,\lambda_{m}\geq0$
\end_inset

 son los valores propios de 
\begin_inset Formula $A^{*}A$
\end_inset

 y 
\begin_inset Formula $E_{1},\dots,E_{m}$
\end_inset

 son los subespacios propios asociados, 
\begin_inset Formula $\rho(A^{*}A)=\max\{\lambda_{1},\dots,\lambda_{m}\}=\max_{k=1}^{m}\max\{R_{A^{*}A}(v)\mid v\in E_{k}\setminus\{0\}\}=\max\{R_{A^{*}A}(v)\mid v\neq0\}$
\end_inset

, y como 
\begin_inset Formula 
\[
R_{A^{*}A}(v)=\frac{\langle Av,v\rangle}{\langle v,v\rangle}=\left\langle A\frac{v}{\sqrt{\langle v,v\rangle}},\frac{v}{\sqrt{\langle v,v\rangle}}\right\rangle =\left\langle A\frac{v}{\Vert v\Vert_{2}},\frac{v}{\Vert v\Vert_{2}}\right\rangle ,
\]

\end_inset

queda 
\begin_inset Formula $\rho(A^{*}A)=\max\{R_{A^{*}A}(v)\mid v\neq0\}=\max\{R_{A^{*}A}(v)\mid\Vert v\Vert_{2}=1\}=\Vert A\Vert_{2}^{2}$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Enumerate
\begin_inset Formula $\Vert A\Vert_{\infty}=\max_{i}\sum_{j}|a_{ij}|$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula 
\begin{align*}
\Vert A\Vert_{\infty} & =\sup\{\Vert Ax\Vert_{\infty}\mid\Vert x\Vert_{\infty}=1\}=\sup\{\max_{k}|Ax|_{k}\mid\max_{k}|x_{k}|=1\}=\\
 & =\sup\left\{ \max_{k}\left|\sum_{i}a_{ki}x_{i}\right|\;\middle|\;\max_{i}|x_{i}|=1\right\} =\max_{k}\sup\left\{ \left|\sum_{i}a_{ki}x_{i}\right|\;\middle|\;\max_{i}|x_{i}|=1\right\} .
\end{align*}

\end_inset

 Este supremo se alcanza cuando, para cada 
\begin_inset Formula $i$
\end_inset

, 
\begin_inset Formula $x_{i}=1$
\end_inset

 si 
\begin_inset Formula $a_{ki}>0$
\end_inset

 o 
\begin_inset Formula $x_{i}=-1$
\end_inset

 si 
\begin_inset Formula $a_{ki}<0$
\end_inset

, con lo que 
\begin_inset Formula $\sup\{|\sum_{i}a_{ki}x_{i}|\mid\max_{i}|x_{i}|=1\}=\left|\sum_{i}|a_{ki}|\right|=\sum_{i}|a_{ki}|$
\end_inset

, luego 
\begin_inset Formula $\Vert A\Vert_{\infty}=\max_{k}\sum_{i}|a_{ki}|$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Enumerate
Si 
\begin_inset Formula $A$
\end_inset

 es normal, 
\begin_inset Formula $\Vert A\Vert_{2}=\rho(A)$
\end_inset

.
\end_layout

\begin_layout Standard
La 
\series bold
norma euclídea
\series default
, 
\begin_inset Formula $\Vert A\Vert_{E}\coloneqq \sqrt{\sum_{i,j}|a_{ij}|^{2}}$
\end_inset

, es una norma matricial no subordinada a ninguna norma en 
\begin_inset Formula $\mathbb{K}^{n}$
\end_inset

, pero es más fácil de calcular que 
\begin_inset Formula $\Vert\cdot\Vert_{2}$
\end_inset

, y 
\begin_inset Formula $\Vert A\Vert_{2}\leq\Vert A\Vert_{E}\leq\sqrt{n}\Vert A\Vert_{2}$
\end_inset

.
\end_layout

\begin_layout Standard
Si 
\begin_inset Formula $A\in{\cal M}_{n}(\mathbb{K})$
\end_inset

:
\end_layout

\begin_layout Enumerate
Toda norma matricial 
\begin_inset Formula $\Vert\cdot\Vert$
\end_inset

 en 
\begin_inset Formula ${\cal M}_{n}(\mathbb{K})$
\end_inset

 cumple 
\begin_inset Formula $\rho(A)\leq\Vert A\Vert$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Sean 
\begin_inset Formula $\lambda$
\end_inset

 un valor propio tal que 
\begin_inset Formula $|\lambda|=\rho(A)$
\end_inset

, 
\begin_inset Formula $p\neq0$
\end_inset

 un vector propio de 
\begin_inset Formula $\lambda$
\end_inset

 y 
\begin_inset Formula $q\in\mathbb{K}^{n}$
\end_inset

 tal que la matriz 
\begin_inset Formula $pq^{t}\neq0$
\end_inset

.
 Entonces 
\begin_inset Formula $\rho(A)\Vert pq^{t}\Vert=\Vert\lambda pq^{t}\Vert=\Vert(Ap)q^{t}\Vert=\Vert A(pq^{t})\Vert\leq\Vert A\Vert\Vert pq^{t}\Vert$
\end_inset

, y despejando, 
\begin_inset Formula $\rho(A)\leq\Vert A\Vert$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Enumerate
Para todo 
\begin_inset Formula $\varepsilon>0$
\end_inset

 existe una norma matricial subordinada 
\begin_inset Formula $\Vert\cdot\Vert$
\end_inset

 tal que 
\begin_inset Formula $\Vert A\Vert\leq\rho(A)+\varepsilon$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Sea 
\begin_inset Formula $U$
\end_inset

 la matriz unitaria tal que 
\begin_inset Formula $U^{-1}AU$
\end_inset

 es triangular superior.
 Entonces la diagonal está formada por los valores propios 
\begin_inset Formula $\lambda_{1},\dots,\lambda_{n}$
\end_inset

, no necesariamente distintos, de 
\begin_inset Formula $A$
\end_inset

.
 
\end_layout

\begin_layout Standard
Sea 
\begin_inset Formula $D_{\delta}\coloneqq \text{diag}(1,\delta,\dots,\delta^{n-1})$
\end_inset

 para 
\begin_inset Formula $\delta>0$
\end_inset

, entonces 
\begin_inset Formula $(UD_{\delta})^{-1}A(UD_{\delta})=D_{\delta}^{-1}U^{-1}AUD_{\delta}=D_{\delta^{-1}}U^{-1}AUD_{\delta}$
\end_inset

, pero 
\begin_inset Formula $(a_{ij})D_{\delta}=(\delta^{j}a_{ij})$
\end_inset

 y 
\begin_inset Formula $D_{\delta^{-1}}(a_{ij})=(\delta^{-i}a_{ij})$
\end_inset

, luego si 
\begin_inset Formula $U^{-1}AU=(u_{ij})$
\end_inset

, 
\begin_inset Formula $D_{\delta}^{-1}U^{-1}AUD_{\delta}=D_{\delta^{-1}}(u_{ij})D_{\delta}=D_{\delta^{-1}}(\delta^{j}u_{ij})=(\delta^{j-i}u_{ij})$
\end_inset

.
 
\end_layout

\begin_layout Standard
La diagonal no cambia, la matriz sigue siendo triangular superior y, para
 
\begin_inset Formula $\delta$
\end_inset

 suficientemente pequeño, 
\begin_inset Formula $\sum_{j=i+1}^{n}|\delta^{j-i}u_{ij}|<\varepsilon$
\end_inset

 para cada 
\begin_inset Formula $i$
\end_inset

.
 Así, 
\begin_inset Formula $\Vert(\delta^{j-i}u_{ij})\Vert_{\infty}=\max_{i}\sum_{j}\delta^{j-i}u_{ij}=\max_{i}(\lambda_{i}+\sum_{j=i+1}^{n}\delta^{j-i}u_{ij})\leq\rho(A)+\varepsilon$
\end_inset

.
 Tomando la norma 
\begin_inset Formula $\Vert v\Vert_{*}\coloneqq \Vert(UD_{\delta})^{-1}v\Vert_{\infty}$
\end_inset

, la norma subordinada a esta cumple 
\begin_inset Formula $\Vert A\Vert_{*}=\Vert(UD_{\delta})^{-1}A(UD_{\delta})\Vert_{\infty}\leq\rho(A)+\varepsilon$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Standard
De aquí que 
\begin_inset Formula $\rho(A)=\inf\{\Vert A\Vert\mid\Vert\cdot\Vert\text{ es una norma matricial en }{\cal M}_{n}(\mathbb{K})\}$
\end_inset

.
\end_layout

\begin_layout Standard
Sea 
\begin_inset Formula $B\in{\cal M}_{n}$
\end_inset

, 
\begin_inset Formula $\lim_{k}B^{k}=0$
\end_inset

 si y sólo si 
\begin_inset Formula $\forall v\in\mathbb{K}^{n},\lim_{k}B^{k}v=0$
\end_inset

, si y sólo si 
\begin_inset Formula $\rho(B)<1$
\end_inset

, si y sólo si existe una norma subordinada tal que 
\begin_inset Formula $\Vert B\Vert<1$
\end_inset

.
\end_layout

\begin_layout Description
\begin_inset Formula $[1\implies2]$
\end_inset

 
\begin_inset Formula $0\leq\lim_{k}\Vert B^{k}v\Vert\leq\lim_{k}\Vert B^{k}\Vert\Vert v\Vert=0\Vert v\Vert=0$
\end_inset

, luego 
\begin_inset Formula $\lim_{k}B^{k}v=0$
\end_inset

.
\end_layout

\begin_layout Description
\begin_inset Formula $[2\implies3]$
\end_inset

 Sea 
\begin_inset Formula $\lambda$
\end_inset

 un valor propio de 
\begin_inset Formula $A$
\end_inset

 y 
\begin_inset Formula $p$
\end_inset

 un vector propio asociado, entonces 
\begin_inset Formula $\lim_{k}B^{k}p=\lim_{k}\lambda^{k}p=p\lim_{k}\lambda^{k}=0$
\end_inset

, luego 
\begin_inset Formula $|\lambda|<1$
\end_inset

.
\end_layout

\begin_layout Description
\begin_inset Formula $[3\implies4]$
\end_inset

 Por el teorema anterior, existe 
\begin_inset Formula $\Vert\cdot\Vert$
\end_inset

 tal que 
\begin_inset Formula $\Vert B\Vert<\rho(B)+(1-\rho(B))=1$
\end_inset

.
\end_layout

\begin_layout Description
\begin_inset Formula $[4\implies1]$
\end_inset

 Sea 
\begin_inset Formula $\Vert\cdot\Vert$
\end_inset

 esta norma, 
\begin_inset Formula $0\leq\lim_{k}\Vert B^{k}\Vert\leq\lim_{k}\Vert B\Vert^{k}=0$
\end_inset

, luego 
\begin_inset Formula $\lim_{k}B^{k}=0$
\end_inset

.
\end_layout

\begin_layout Standard
Toda norma matricial cumple 
\begin_inset Formula $\lim_{k}\Vert B^{k}\Vert^{1/k}=\rho(B)$
\end_inset

.
 
\series bold
Demostración:
\series default
 Sabemos que 
\begin_inset Formula $\rho(B)\leq\Vert B\Vert$
\end_inset

, y como 
\begin_inset Formula $\rho(B)=\rho(B^{k})^{1/k}$
\end_inset

, queda 
\begin_inset Formula $\rho(B)\leq\Vert B^{k}\Vert^{1/k}$
\end_inset

 para todo 
\begin_inset Formula $k$
\end_inset

.
 Fijado ahora 
\begin_inset Formula $\varepsilon>0$
\end_inset

, sea 
\begin_inset Formula $B_{\varepsilon}\coloneqq \frac{B}{\rho(B)+\varepsilon}$
\end_inset

, se tiene 
\begin_inset Formula $\rho(B_{\varepsilon})<1$
\end_inset

, por lo que 
\begin_inset Formula $\lim_{k}B_{\varepsilon}^{k}=0$
\end_inset

 y existe 
\begin_inset Formula $k_{0}$
\end_inset

 tal que, para 
\begin_inset Formula $k\geq k_{0}$
\end_inset

, 
\begin_inset Formula $B_{\varepsilon}^{k}\leq1$
\end_inset

, pero entonces 
\begin_inset Formula $\Vert B_{\varepsilon}^{k}\Vert=\frac{\Vert B^{k}\Vert}{(\rho(B)+\varepsilon)^{k}}\leq1$
\end_inset

 y 
\begin_inset Formula $\Vert B^{k}\Vert^{1/k}\leq\rho(B)+\varepsilon$
\end_inset

.
\end_layout

\begin_layout Section
Análisis del error
\end_layout

\begin_layout Standard
Sean 
\begin_inset Formula $A\in{\cal M}_{m\times n}$
\end_inset

 invertible, 
\begin_inset Formula $b\in\mathbb{K}^{n}\setminus0$
\end_inset

 y 
\begin_inset Formula $\Vert\cdot\Vert$
\end_inset

 una norma subordinada:
\end_layout

\begin_layout Enumerate
Considerando los sistemas 
\begin_inset Formula $Ax=b$
\end_inset

 y 
\begin_inset Formula $A(x+\Delta x)=b+\Delta b$
\end_inset

, 
\begin_inset Formula $\frac{\Vert\Delta x\Vert}{\Vert x\Vert}\leq\Vert A\Vert\Vert A^{-1}\Vert\frac{\Vert\Delta b\Vert}{\Vert b\Vert}$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
Es claro que 
\begin_inset Formula $A\Delta x=\Delta b$
\end_inset

 y por tanto 
\begin_inset Formula $\Delta x=A^{-1}\Delta b$
\end_inset

, con lo que 
\begin_inset Formula $\Vert\Delta x\Vert\leq\Vert A^{-1}\Vert\Vert\Delta b\Vert$
\end_inset

, y como también 
\begin_inset Formula $\Vert b\Vert=\Vert Ax\Vert\leq\Vert A\Vert\Vert x\Vert$
\end_inset

, podemos obtener la fórmula despejando.
\end_layout

\end_deeper
\begin_layout Enumerate
Considerando los sistemas 
\begin_inset Formula $Ax=b$
\end_inset

 y 
\begin_inset Formula $(A+\Delta A)(x+\Delta x)=b$
\end_inset

, 
\begin_inset Formula $\frac{\Vert\Delta x\Vert}{\Vert x+\Delta x\Vert}\leq\Vert A^{-1}\Vert\Vert\Delta A\Vert$
\end_inset

 y 
\begin_inset Formula $\frac{\Vert\Delta x\Vert}{\Vert x\Vert}\leq\frac{\Vert A^{-1}\Vert\Vert\Delta A\Vert}{1-\Vert A^{-1}\Vert\Vert\Delta A\Vert}$
\end_inset

.
\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula $(A+\Delta A)(x+\Delta x)=Ax+A\Delta x+\Delta A(x+\Delta x)=b+A\Delta x+\Delta A(x+\Delta x)=b$
\end_inset

, luego 
\begin_inset Formula $A\Delta x=-\Delta A(x+\Delta x)$
\end_inset

 y por tanto 
\begin_inset Formula $\Delta x=-A^{-1}\Delta A(x+\Delta x)$
\end_inset

.
 Entonces 
\begin_inset Formula $\Vert\Delta x\Vert\leq\Vert A^{-1}\Vert\Vert\Delta A\Vert\Vert x+\Delta x\Vert$
\end_inset

, lo que nos da la primera desigualdad.
 A partir de aquí, 
\begin_inset Formula $\Vert x+\Delta x\Vert\leq\Vert x\Vert+\Vert\Delta x\Vert\leq\Vert x\Vert+\Vert A^{-1}\Vert\Vert\Delta A\Vert\Vert x+\Delta x\Vert$
\end_inset

 y por tanto 
\begin_inset Formula $\Vert x+\Delta x\Vert(1-\Vert A^{-1}\Vert\Vert\Delta A\Vert)\leq\Vert x\Vert$
\end_inset

, y despejando de esto y la primera desigualdad se obtiene la segunda.
\end_layout

\end_deeper
\begin_layout Standard
Llamamos 
\series bold
número de condición
\series default
 de 
\begin_inset Formula $A$
\end_inset

 respecto a la norma 
\begin_inset Formula $\Vert\cdot\Vert$
\end_inset

 a 
\begin_inset Formula $\text{cond}A\coloneqq \Vert A\Vert\Vert A^{-1}\Vert$
\end_inset

, con lo que si 
\begin_inset Formula $Ax=b$
\end_inset

 y 
\begin_inset Formula $A(x+\Delta x)=b+\Delta b$
\end_inset

 entonces 
\begin_inset Formula $\frac{\Vert\Delta x\Vert}{\Vert x\Vert}\leq\text{cond}A\frac{\Vert\Delta b\Vert}{\Vert b\Vert}$
\end_inset

, y si 
\begin_inset Formula $Ax=b$
\end_inset

 y 
\begin_inset Formula $(A+\Delta)(x+\Delta x)=b$
\end_inset

 entonces 
\begin_inset Formula $\frac{\Vert\Delta x\Vert}{\Vert x+\Delta x\Vert}\leq\text{cond}A\frac{\Vert\Delta A\Vert}{\Vert A\Vert}$
\end_inset

.
 Estas desigualdades son las mejores posibles en el sentido de que se pueden
 encontrar 
\begin_inset Formula $b,\Delta b\neq0$
\end_inset

 para los que se obtiene la igualdad en la primera desigualdad y 
\begin_inset Formula $b\neq0$
\end_inset

 y 
\begin_inset Formula $\Delta A\neq0$
\end_inset

 para los que se obtiene en la segunda.
\end_layout

\begin_layout Standard
Llamamos 
\begin_inset Formula $\text{cond}_{p}(A)\coloneqq \Vert A^{-1}\Vert_{p}\Vert A\Vert_{p}$
\end_inset

.
 Para toda 
\begin_inset Formula $A\in{\cal M}_{n}$
\end_inset

 invertible:
\end_layout

\begin_layout Enumerate
\begin_inset Formula $\text{cond}A\geq1$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Formula $\text{cond}A=\text{cond}A^{-1}$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Formula $\forall\alpha\in\mathbb{K}\setminus0,\text{cond}(\alpha A)=\text{cond}A$
\end_inset

.
\end_layout

\begin_layout Enumerate
Sean 
\begin_inset Formula $M$
\end_inset

 el mayor valor singular de 
\begin_inset Formula $A$
\end_inset

 y 
\begin_inset Formula $m$
\end_inset

 el menor, 
\begin_inset Formula $\text{cond}_{2}A=\frac{M}{m}$
\end_inset

.
\end_layout

\begin_layout Enumerate
Si 
\begin_inset Formula $A$
\end_inset

 es normal, sean 
\begin_inset Formula $M$
\end_inset

 el mayor valor propio de 
\begin_inset Formula $A$
\end_inset

 y 
\begin_inset Formula $m$
\end_inset

 el menor, 
\begin_inset Formula $\text{cond}_{2}A=\rho(A)\rho(A^{-1})=\frac{|\lambda_{n}(A)|}{|\lambda_{1}(A)|}$
\end_inset

.
\end_layout

\begin_layout Enumerate
Sea 
\begin_inset Formula $U$
\end_inset

 una matriz unitaria, 
\begin_inset Formula $\text{cond}_{2}A=\text{cond}_{2}(UA)=\text{cond}_{2}(AU)=\text{cond}_{2}(U^{-1}AU)$
\end_inset

.
\end_layout

\begin_layout Standard
Sean 
\begin_inset Formula $A$
\end_inset

 diagonalizable, 
\begin_inset Formula $P$
\end_inset

 invertible con 
\begin_inset Formula $D\coloneqq P^{-1}AP=:\text{diag}(\lambda_{i})$
\end_inset

, 
\begin_inset Formula $\Vert\cdot\Vert$
\end_inset

 una norma con 
\begin_inset Formula $\Vert\text{diag}(d_{1},\dots,d_{n})\Vert=\max_{i}|d_{i}|$
\end_inset

 para toda matriz diagonal y 
\begin_inset Formula $D_{i}\coloneqq B(\lambda_{i},\text{cond}(P)\Vert\Delta A\Vert)\subseteq\mathbb{C}$
\end_inset

,
\begin_inset Formula 
\[
\sigma(A+\Delta A)\subseteq\bigcup_{i=1}^{n}D_{i}.
\]

\end_inset


\end_layout

\begin_layout Standard

\series bold
Demostración:
\series default
 Sea 
\begin_inset Formula $\lambda\neq\lambda_{1},\dots,\lambda_{n}$
\end_inset

, 
\begin_inset Formula $D-\lambda I$
\end_inset

 es invertible con inversa 
\begin_inset Formula $\text{diag}(\frac{1}{\lambda_{1}-\lambda},\dots,\frac{1}{\lambda_{n}-\lambda})$
\end_inset

.
 Si 
\begin_inset Formula $\lambda$
\end_inset

 es valor propio de 
\begin_inset Formula $A+\Delta A$
\end_inset

, 
\begin_inset Formula $A+\Delta A-\lambda I$
\end_inset

 no debe tener inversa, por lo que tampoco debe tener inversa 
\begin_inset Formula $P^{-1}(A+\Delta A-\lambda I)P=P^{-1}AP-P^{-1}\lambda IP+P^{-1}\Delta AP=D-\lambda I+P^{-1}\Delta AP=(D-\lambda I)(I+(D-\lambda I)^{-1}P^{-1}\Delta AP)=:(D-\lambda I)(I+B)$
\end_inset

, con lo que 
\begin_inset Formula $I+B$
\end_inset

 no debe ser invertible.
 Ahora bien, si 
\begin_inset Formula $\Vert B\Vert<1$
\end_inset

, 
\begin_inset Formula 
\begin{multline*}
(I+B)\sum_{k=0}^{n}(-1)^{k}B^{k}=\sum_{k=0}^{n}(-1)^{k}(B^{k}+B^{k+1})=\sum_{k=0}^{n}(-1)^{k}B^{k}+\sum_{k=1}^{n+1}(-1)^{k-1}B^{n}=\\
=\sum_{k=0}^{n}(-1)^{k}B^{k}-\sum_{k=1}^{n+1}(-1)^{k}B^{k}=B^{0}+(-1)^{n}B^{n+1}=I+(-1)^{n}B^{n+1},
\end{multline*}

\end_inset

 pero 
\begin_inset Formula $\lim_{n}B^{n+1}=0$
\end_inset

, luego 
\begin_inset Formula 
\[
(I+B)\sum_{k=0}^{\infty}(-1)^{k}B^{k}=\lim_{n}(I+(-1)^{n}B^{n+1})=I
\]

\end_inset

 y 
\begin_inset Formula $\sum_{k=0}^{\infty}(-1)^{k}B^{k}$
\end_inset

 es la inversa de 
\begin_inset Formula $I+B$
\end_inset

.
 Por tanto 
\begin_inset Formula $\Vert B\Vert\geq1$
\end_inset

, luego 
\begin_inset Formula $1\leq\Vert(D-\lambda I)^{-1}P^{-1}\Delta AP\Vert\leq\Vert(D-\lambda I)^{-1}\Vert\Vert P^{-1}\Vert\Vert\Delta A\Vert\Vert P\Vert=\Vert(D-\lambda I)^{-1}\Vert\Vert\Delta A\Vert\text{cond}(P)$
\end_inset

, y como 
\begin_inset Formula 
\[
\Vert(D-\lambda I)^{-1}\Vert=\max_{k}\left|\frac{1}{\lambda_{k}-\lambda}\right|=\frac{1}{\min_{k}|\lambda_{k}-\lambda|},
\]

\end_inset

 queda 
\begin_inset Formula $1\leq\frac{\Vert\Delta A\Vert\text{cond}(P)}{\min_{k}|\lambda_{k}-\lambda|}$
\end_inset

 y por tanto 
\begin_inset Formula $\min_{k}|\lambda_{k}-\lambda|\leq\Vert\Delta A\Vert\text{cond}(P)$
\end_inset

.
 
\end_layout

\end_body
\end_document