diff options
Diffstat (limited to 'anm/n1.lyx')
| -rw-r--r-- | anm/n1.lyx | 310 |
1 files changed, 98 insertions, 212 deletions
@@ -278,7 +278,7 @@ traza a \begin_inset Formula \[ -\text{tr}A:=\sum_{k=1}^{n}a_{kk}. +\text{tr}A:=\sum_{k=1}^{n}A_{kk}. \] \end_inset @@ -395,7 +395,7 @@ sistema \begin_inset Formula $n$ \end_inset - incógnitas es un sistema de la forma + incógnitas es uno de la forma \begin_inset Formula \[ \left\{ \begin{aligned}a_{11}x_{1}+\dots+a_{1n}x_{n} & =b_{1},\\ @@ -522,15 +522,15 @@ Una \series bold base \series default - de un espacio vectorial -\begin_inset Formula $E$ + de un +\begin_inset Formula $\mathbb{K}$ \end_inset - de dimensión finita sobre un cuerpo -\begin_inset Formula $\mathbb{K}$ +-espacio vectorial +\begin_inset Formula $E$ \end_inset - es una tupla + de dimensión finita es una tupla \begin_inset Formula $(v_{1},\dots,v_{n})$ \end_inset @@ -563,7 +563,7 @@ con \begin_inset Formula $(x_{1},\dots,x_{n})\in\mathbb{K}^{n}$ \end_inset -, y por tanto con la correspondiente matriz columna. +, y con la correspondiente matriz columna. \end_layout \begin_layout Standard @@ -584,7 +584,7 @@ producto escalar \end_inset bilineal simétrica tal que -\begin_inset Formula $\forall f\in E\setminus\{0\},\langle f,f\rangle>0$ +\begin_inset Formula $\forall f\in E\setminus0,\langle f,f\rangle>0$ \end_inset . @@ -686,169 +686,6 @@ ortogonales \end_inset . - -\end_layout - -\begin_layout Standard -Una -\series bold -norma -\series default - en un -\begin_inset Formula $\mathbb{R}$ -\end_inset - --espacio vectorial -\begin_inset Formula $E$ -\end_inset - - es una aplicación -\begin_inset Formula $\Vert\cdot\Vert:E\to\mathbb{K}$ -\end_inset - - tal que -\begin_inset Formula $\forall v,w\in E,t\in\mathbb{R}:$ -\end_inset - - -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $\Vert tv\Vert=|t|\Vert v\Vert$ -\end_inset - -. -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $\Vert v+w\Vert\leq\Vert v\Vert+\Vert w\Vert$ -\end_inset - -. -\end_layout - -\begin_layout Enumerate -\begin_inset Formula $v\neq0\implies\Vert v\Vert>0$ -\end_inset - -. -\end_layout - -\begin_layout Standard -Llamamos -\series bold -norma euclídea -\series default - en -\begin_inset Formula $\mathbb{R}^{n}$ -\end_inset - - a -\begin_inset Formula $\Vert v\Vert:=\sqrt{\langle v,v\rangle}$ -\end_inset - -. -\end_layout - -\begin_layout Standard -Sean -\begin_inset Formula $f:V\to W$ -\end_inset - - una aplicación lineal y -\begin_inset Formula ${\cal B}:=(v_{1},\dots,v_{n})$ -\end_inset - - y -\begin_inset Formula ${\cal B}':=(w_{1},\dots,w_{m})$ -\end_inset - - bases respectivas de -\begin_inset Formula $V$ -\end_inset - - y -\begin_inset Formula $W$ -\end_inset - -, si -\begin_inset Formula -\[ -\left\{ \begin{aligned}f(v_{1}) & =a_{11}w_{1}+\dots+a_{m1}w_{1},\\ - & \vdots\\ -f(v_{m}) & =a_{1n}w_{1}+\dots+a_{mn}w_{m}, -\end{aligned} -\right. -\] - -\end_inset - -llamamos -\series bold -matriz asociada -\series default - a -\begin_inset Formula $f$ -\end_inset - - con respecto de las bases -\begin_inset Formula ${\cal B}$ -\end_inset - - y -\begin_inset Formula ${\cal B}'$ -\end_inset - - a -\begin_inset Formula $(a_{ij})_{1\leq i\leq m}^{1\leq j\leq n}$ -\end_inset - -. - Dadas dos aplicaciones lineales -\begin_inset Formula $U\overset{f}{\to}V\overset{g}{\to}W$ -\end_inset - -, -\begin_inset Formula $g\circ f$ -\end_inset - - también es lineal, y si -\begin_inset Formula $U$ -\end_inset - -, -\begin_inset Formula $V$ -\end_inset - - y -\begin_inset Formula $W$ -\end_inset - - son de dimensión finita y -\begin_inset Formula $f$ -\end_inset - - y -\begin_inset Formula $g$ -\end_inset - - tienen matrices respectivas -\begin_inset Formula $A$ -\end_inset - - y -\begin_inset Formula $B$ -\end_inset - -, -\begin_inset Formula $g\circ f$ -\end_inset - - tiene matriz -\begin_inset Formula $BA$ -\end_inset - - respecto de las mismas bases. \end_layout \begin_layout Section @@ -1443,11 +1280,11 @@ El cociente de Rayleigh \series default de una matriz -\begin_inset Formula $A\in{\cal M}_{n}$ +\begin_inset Formula $A\in{\cal M}_{n}(\mathbb{C})$ \end_inset es una aplicación -\begin_inset Formula $R_{A}:\mathbb{C}^{n}\setminus\{0\}\to\mathbb{C}$ +\begin_inset Formula $R_{A}:\mathbb{C}^{n}\setminus0\to\mathbb{C}$ \end_inset dada por @@ -1474,7 +1311,7 @@ Sean \begin_inset Formula $A\in{\cal M}_{n}$ \end_inset - es hermitiana con valores propios + hermitiana con valores propios \begin_inset Formula $\lambda_{1}\leq\dots\leq\lambda_{n}$ \end_inset @@ -1498,7 +1335,11 @@ Sean \begin_inset Formula $k$ \end_inset -, + ( +\begin_inset Formula $E_{0}=\{0\}$ +\end_inset + +) y \begin_inset Formula ${\cal S}_{k}$ \end_inset @@ -1510,14 +1351,6 @@ Sean \begin_inset Formula $k$ \end_inset -, -\begin_inset Formula $E_{0}:=\{0\}$ -\end_inset - - y -\begin_inset Formula ${\cal S}_{k}:=\{E_{0}\}$ -\end_inset - . Entonces, para \begin_inset Formula $1\leq k\leq n$ @@ -1750,11 +1583,15 @@ Sea \begin_inset Formula $E$ \end_inset - un -\begin_inset Formula $\mathbb{K}$ + un espacio vectorial sobre +\begin_inset Formula $\mathbb{R}$ +\end_inset + + o +\begin_inset Formula $\mathbb{C}$ \end_inset --espacio vectorial, una +, una \series bold norma \series default @@ -1892,7 +1729,19 @@ norma matricial \begin_inset Formula ${\cal M}_{n}(\mathbb{K})$ \end_inset - es una que cumple +, 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 @@ -1907,7 +1756,63 @@ norma matricial , llamamos \series bold -norma matricial subordinada +norma matricial sub +\begin_inset ERT +status open + +\begin_layout Plain Layout + +\series bold + +\backslash +- +\end_layout + +\end_inset + +or +\begin_inset ERT +status open + +\begin_layout Plain Layout + +\series bold + +\backslash +- +\end_layout + +\end_inset + +di +\begin_inset ERT +status open + +\begin_layout Plain Layout + +\series bold + +\backslash +- +\end_layout + +\end_inset + +na +\begin_inset ERT +status open + +\begin_layout Plain Layout + +\series bold + +\backslash +- +\end_layout + +\end_inset + +da \series default a la norma \begin_inset Formula $\Vert\cdot\Vert$ @@ -1920,7 +1825,7 @@ norma matricial subordinada dada por \begin_inset Formula \[ -\Vert A\Vert:=\sup\left\{ \frac{\Vert Ax\Vert}{\Vert x\Vert}\right\} _{x\in\mathbb{K}^{n}\setminus\{0\}}=\sup\left\{ \frac{\Vert Ax\Vert}{\Vert x\Vert}\right\} _{\Vert x\Vert\leq1}=\sup\left\{ \Vert Ax\Vert\right\} _{\Vert x\Vert=1}. +\Vert A\Vert:=\sup_{x\in\mathbb{K}^{n}\setminus\{0\}}\frac{\Vert Ax\Vert}{\Vert x\Vert}=\sup_{\Vert x\Vert\leq1}\frac{\Vert Ax\Vert}{\Vert x\Vert}=\sup_{\Vert x\Vert=1}\Vert Ax\Vert. \] \end_inset @@ -1941,13 +1846,6 @@ Entonces, para \end_layout \begin_layout Standard -\begin_inset Newpage pagebreak -\end_inset - - -\end_layout - -\begin_layout Standard Sea \begin_inset Formula $A:=(a_{ij})_{ij}\in{\cal M}_{n}(\mathbb{C})$ \end_inset @@ -2139,7 +2037,7 @@ norma euclídea \begin_inset Formula $\Vert\cdot\Vert_{2}$ \end_inset - y +, y \begin_inset Formula $\Vert A\Vert_{2}\leq\Vert A\Vert_{E}\leq\sqrt{n}\Vert A\Vert_{2}$ \end_inset @@ -2497,7 +2395,7 @@ Sean \end_inset invertible, -\begin_inset Formula $0\neq b\in\mathbb{K}^{n}$ +\begin_inset Formula $b\in\mathbb{K}^{n}\setminus0$ \end_inset y @@ -2682,7 +2580,7 @@ Llamamos \end_layout \begin_layout Enumerate -\begin_inset Formula $\forall\alpha\in\mathbb{K}\setminus\{0\},\text{cond}(\alpha A)=\text{cond}A$ +\begin_inset Formula $\forall\alpha\in\mathbb{K}\setminus0,\text{cond}(\alpha A)=\text{cond}A$ \end_inset . @@ -2733,18 +2631,6 @@ Si \end_layout \begin_layout Enumerate -Si -\begin_inset Formula $A$ -\end_inset - - es unitaria, -\begin_inset Formula $\text{cond}_{2}U=1$ -\end_inset - -. -\end_layout - -\begin_layout Enumerate Sea \begin_inset Formula $U$ \end_inset |