diff options
Diffstat (limited to 'anm/n1.lyx')
| -rw-r--r-- | anm/n1.lyx | 26 |
1 files changed, 15 insertions, 11 deletions
@@ -1519,7 +1519,7 @@ Queremos ver que . Si -\begin_inset Formula $E_{k-1}^{\bot}:=\{v\in V:v\bot E_{k-1}\}$ +\begin_inset Formula $E_{k-1}^{\bot}:=\{v\in V\mid v\bot E_{k-1}\}$ \end_inset , basta ver que para todo subespacio @@ -1860,7 +1860,7 @@ Sea \begin_deeper \begin_layout Standard -\begin_inset Formula $\sup\{\Vert Ax\Vert:\Vert x\Vert=1\}=\sup\{\sum_{k}|Ax|_{k}:\sum_{k}|x_{k}|=1\}=\sup\{\sum_{k,i}|a_{ki}||x_{i}|:\sum_{i}|x_{i}|=1\}$ +\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 . @@ -1889,7 +1889,7 @@ Sea \end_inset luego -\begin_inset Formula $\sup\{\sum_{k,i}|a_{ki}||x_{i}|:\sum_{i}|x_{i}|=1\}=\max_{i}\sum_{k}|a_{ki}|$ +\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 . @@ -1922,10 +1922,14 @@ luego \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}}:\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):\Vert x\Vert_{2}=1\right\} $ +\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 + pero si \begin_inset Formula $\lambda_{1},\dots,\lambda_{m}\geq0$ \end_inset @@ -1938,7 +1942,7 @@ luego \end_inset son los subespacios propios asociados, -\begin_inset Formula $\rho(A^{*}A)=\max\{\lambda_{1},\dots,\lambda_{m}\}=\max_{k=1}^{m}\max\{R_{A^{*}A}(v):v\in E_{k}\setminus\{0\}\}=\max\{R_{A^{*}A}(v):v\neq0\}$ +\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 @@ -1950,7 +1954,7 @@ R_{A^{*}A}(v)=\frac{\langle Av,v\rangle}{\langle v,v\rangle}=\left\langle A\frac \end_inset queda -\begin_inset Formula $\rho(A^{*}A)=\max\{R_{A^{*}A}(v):v\neq0\}=\max\{R_{A^{*}A}(v):\Vert v\Vert_{2}=1\}=\Vert A\Vert_{2}^{2}$ +\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 . @@ -1968,8 +1972,8 @@ queda \begin_layout Standard \begin_inset Formula \begin{align*} -\Vert A\Vert_{\infty} & =\sup\{\Vert Ax\Vert_{\infty}:\Vert x\Vert_{\infty}=1\}=\sup\{\max_{k}|Ax|_{k}:\max_{k}|x_{k}|=1\}=\\ - & =\sup\left\{ \max_{k}\left|\sum_{i}a_{ki}x_{i}\right|:\max_{i}|x_{i}|=1\right\} =\max_{k}\sup\left\{ \left|\sum_{i}a_{ki}x_{i}\right|:\max_{i}|x_{i}|=1\right\} . +\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 @@ -1995,7 +1999,7 @@ queda \end_inset , con lo que -\begin_inset Formula $\sup\{|\sum_{i}a_{ki}x_{i}|:\max_{i}|x_{i}|=1\}=\left|\sum_{i}|a_{ki}|\right|=\sum_{i}|a_{ki}|$ +\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 @@ -2211,7 +2215,7 @@ La diagonal no cambia, la matriz sigue siendo triangular superior y, para \end_deeper \begin_layout Standard De aquí que -\begin_inset Formula $\rho(A)=\inf\{\Vert A\Vert:\Vert\cdot\Vert\text{ es una norma matricial en }{\cal M}_{n}(\mathbb{K})\}$ +\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 . |