# LaTeX2HTML 2002 (1.62)
# Associate images original text with physical files.


$key = q/mathcal{A}=(a_{ij});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="$\mathcal{A}=(a_{ij})$">|; 

$key = q/lambda_1,ldots,lambda_n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$\lambda_1,\ldots,\lambda_n$">|; 

$key = q/x,yinmathbfR^n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$x,y\in \mathbf R^n$">|; 

$key = q/v;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$v$">|; 

$key = q/v=(v_1,ldots,v_n);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$v=(v_1,\ldots,v_n)$">|; 

$key = q/lambda_1,ldots,lambda_ninmathbf{R};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$\lambda_1,\ldots,\lambda_n\in\mathbf{R}$">|; 

$key = q/{eqnarraystar}Q_mathcalA(x)&=&H_mathcalA(x,x)=x^tmathcal{A}x=<x,mathcal{A}x>&=&sum_ilambda_ixi_i^2{eqnarraystar};MSF=1.6;LFS=11;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="318" HEIGHT="74" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="\begin{eqnarray*}
Q_\mathcal A(x) &amp;=&amp;
H_\mathcal A(x,x)=x^t\mathcal{A}x=&lt;x,\mathcal{A}x&gt;\\\\
&amp;=&amp; \sum_i\lambda_i \xi_i^2
\end{eqnarray*}">|; 

$key = q/y=displaystylesum_{i=1}^{n}eta_iv_i;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$y=\displaystyle\sum_{i=1}^{n} \eta_iv_i$">|; 

$key = q/mathcal{A}v_i=lambda_iv_i;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$\mathcal{A}v_i=\lambda_iv_i$">|; 

$key = q/xi=(xi_1,ldots,xi_n),,eta=(eta_1,ldots,eta_n)inmathbfR^n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$\xi=(\xi_1,\ldots,\xi_n),  \eta=(\eta_1,\ldots,\eta_n)\in\mathbf
R^n$">|; 

$key = q/H_mathcalA;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$H_\mathcal A$">|; 

$key = q/mathcal{A};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$\mathcal{A}$">|; 

$key = q/{displaymath}H_mathcal{A}(v_i,v_j)=<v_i,mathcal{A}v_j>=lambda_j<v_i,v_j>=left{talar{{lcr}0&se&ineqjlambda_j&se&i=jtabular{right.{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="444" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="\begin{displaymath}
H_\mathcal{A}(v_i,v_j)=&lt;v_i,\mathcal{A}v_j&gt;=\lambda_j&lt;v_i,v...
...$\neq j$\\\\
$\lambda_j$ &amp; se &amp; $i=j$
\end{tabular}
\right.
\end{displaymath}">|; 

$key = q/i=1ldotsn;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$i=1\ldots n$">|; 

$key = q/{eqnarraystar}H_mathcalA(x,y)&=&<x,mathcalAy>=<sum_ixi_iv_i,mathcalAsum_ieta_iv_ieta_i<v_i,v_j>&=&sum_ilambda_ixi_ieta_i{eqnarraystar};MSF=1.6;LFS=11;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="419" HEIGHT="137" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="\begin{eqnarray*}
H_\mathcal A(x,y)&amp;=&amp; &lt;x,\mathcal Ay&gt;=&lt;\sum_i\xi_iv_i,\mathcal...
...lambda_i\xi_i\eta_i&lt;v_i,v_j&gt;\\\\
&amp;=&amp;\sum_i\lambda_i\xi_i\eta_i
\end{eqnarray*}">|; 

$key = q/{displaymath}{mathcalD}=left(array{{cccc}lambda_1&0&cdots&00&lambda_2&cdots&0vdovdots&ddots&vdots0&0&cdots&lambda_narray{right).{displaymath};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="102" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="\begin{displaymath}
{\mathcal D} = \left(
\begin{array}{cccc}
\lambda_1 &amp; 0 &amp;...
...\vdots \\\\
0 &amp; 0 &amp; \cdots &amp; \lambda_n
\end{array}
\right).
\end{displaymath}">|; 

$key = q/H_mathcal{A}(x,y)=x^tmathcal{A}y=<x,mathcal{A}y>;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$H_\mathcal{A}(x,y)=x^t\mathcal{A}y=&lt;x,\mathcal{A}y&gt;$">|; 

$key = q/forall,,x,yinmathbf{R}^n;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$\forall   x,y\in\mathbf{R}^n$">|; 

$key = q/H_mathcal{A}:mathbf{R}^ntimesmathbf{R}^nrightarrowmathbf{R};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$H_\mathcal{A}:\mathbf{R}^n\times \mathbf{R}^n\rightarrow \mathbf{R}$">|; 

$key = q/Q_mathcalA;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$Q_\mathcal A$">|; 

$key = q/x,y;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$x,y$">|; 

$key = q/x=displaystylesum_{i=1}^{n}xi_iv_i;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$x=\displaystyle\sum_{i=1}^{n} \xi_iv_i$">|; 

$key = q/v_1,ldots,v_ninmathbf{R^n};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$v_1,\ldots,v_n\in\mathbf{R^n}$">|; 

$key = q/ntimesn;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$n\times n$">|; 

$key = q/mathcalA;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$\mathcal A$">|; 

$key = q/Q_mathcal{A}:mathbf{R}^nrightarrowmathbf{R};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="$Q_\mathcal{A}:\mathbf{R}^n\rightarrow \mathbf{R}$">|; 

1;

