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\font\fifteenbxti=cmbxti10 at 15pt
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{\newpage
\clearpage
\samepage \begin{eqnarray*}\underline{e}_1 & = & ( 1, 0, 0, \ldots 0 ) \\ 
\underline{e}_2 & = & ( 0, 1, \ldots 0 ) \\ 
\underline{e}_n & = & ( 0, 0, \ldots 1 )
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
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\samepage \setbox\sizebox=\hbox{$\underline{x} = \left( x_1, x_2, \ldots x_n
\right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline401: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \begin{displaymath}\underline{x} = \sum_k x_k \underline{e}_k \end{displaymath}
}

{\newpage
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{\newpage
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\underline{e}_k \rbrace$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline405: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\underline{y} & = & A \underline{x} \\ 
\underline{y} & = & \sum_l y_l\, \underline{e}_l \\ 
\underline{x} & = & \sum_k x_k\, \underline{e}_k \\ 
\hbox{Components are}\ y_l & = & \sum_k A_{lk}\, x_k
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\underline{x} = |x > \hbox{called a ket} \end{displaymath}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\underline{x}^T {\fifteenbxti A} \underline{x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline409: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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{\newpage
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\samepage \setbox\sizebox=\hbox{$\langle y \vert$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline413: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$\langle y \vert x \rangle$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline415: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}A_{lk} & = & \underline{e}_l^T\, {\fifteenbxti A}\, \underline{e}_k \\ 
& = & \langle e_l \vert A \vert e_k \rangle
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\lbrace
\underline{e}_k \rbrace$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline419: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\lbrace e_k \rbrace$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline421: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\langle \underline{e}_k \vert \underline{e}_l \rangle & = &
\delta_{lk} \\ 
& = & 0 \hspace{2em} l \neq k \\ 
& = & 1 \hspace{2em} l = k
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\delta_{lk} = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline423: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A^T = \mbox{Transpose} \hspace{2em} \left( A^T \right)_{ij} =
A_{ji}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline429: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A^* = \mbox{Complex\ Conjugate} \hspace{1em} \left( A^*
\right)_{ij} = A^*_{ij}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline431: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A^+ = \mbox{Hermitean Conjugate} \hspace{1em} \left( A^+
\right)_{ij} = A^*_{ji}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline433: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A = A^+$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline435: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A \underline{v} = \lambda \underline{v}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline437: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A \vert v > = \lambda \vert
v >$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline441: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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}

{\newpage
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\samepage \setbox\sizebox=\hbox{$\lambda$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline447: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\underline{e}_1 = ( 1, 0, 0 \ldots 0 )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline449: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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}

{\newpage
\clearpage
\samepage \begin{eqnarray*}A \underline{e}_1 = \lambda_1 \underline{e}_1 & = & \sum_l \left(
\lambda_1 \delta_{l1} \right) \underline{e}_l \\ 
\lambda_1 \delta_{l1} & = & \sum_k A_{l1} \delta_{k1} \\ 
A & = & \left\lbrack \begin{array}{l} 
\lambda_1 \\ 
0 \\ 
0 \hspace{2em} \mbox{ANY} \\ 
0 \hspace{2em} \mbox{VALUE} \\ 
0 \\ 
\vdots \end{array} \right\rbrack \\ 
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}A = \left\lbrack \begin{array}{ccccccc}
\lambda_1 & 0 & 0 & 0 & 0 & \ldots & 0 \\ 
0 \\ 
0 \\ 
0 && A&N&Y \\ 
0 && V&A&L&U&E \\ 
0 \end{array} \right\rbrack \\ 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\underline{e}_2 \ldots \underline{e}_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline461: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$\lambda_2 \ldots \lambda_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline463: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\underline{e}_2 & = & ( 0, 1, 0, \ldots 0 ) \\ 
\underline{e}_n & = & ( 0, 0, \ldots 1 ) 
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{displaymath}A = \left\lbrack \begin{array}{ccccc}
\lambda_1 \\ 
& \lambda_2 && 0 \\ 
&& \lambda_3 \\ 
& 0 && \ddots \\ 
&&&& \lambda_n \end{array} \right\rbrack
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\lambda_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline467: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A\underline{v} = \lambda \underline{v}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline471: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$( A - \lambda I ) \underline{v} = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline473: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
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}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$B = A - \lambda I$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline479: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}B \underline{v} = 0 & \mbox{and} & \underline{v} \neq \left\lbrack
0, \ldots 0 \right\rbrack \\ 
b_{11} v_1 + b_{12} v_2 & \ldots & + b_{1n} v_n = 0 \\ 
\ \ \vdots \\ 
b_{n1} v_1 \ldots & \ldots & b_{nn} v_n = 0
\end{eqnarray*}
}

{\newpage
\clearpage
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}

{\newpage
\clearpage
\samepage \begin{displaymath}v_1 = v_2 = \ldots = v_k = \ldots = v_n = 0
\end{displaymath}
}

\stepcounter{subsection}
{\newpage
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{\newpage
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(B)=0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline491: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\bullet\ n=2 \hspace{3em} \mbox{det}\ B & = & b_{11}\, b_{22} -
b_{12} b_{21} \\ 
\bullet\ n=3 \hspace{3em} \mbox{det}\ B & = & b_{11}\, \mbox{det}\
\left\lbrack \begin{array}{cc}
b_{22} & b_{23} \\ 
b_{32} & b_{33} \end{array} \right\rbrack \\ 
& - & b_{12}\, \mbox{det}\ \left\lbrack \begin{array}{cc}
b_{21} & b_{23} \\ 
b_{31} & b_{33} \end{array} \right\rbrack \\ 
& + & b_{13}\, \mbox{det}\ \left\lbrack \begin{array}{cc}
b_{22} & b_{23} \\ 
b_{32} & b_{33} \end{array} \right\rbrack \\ 
\bullet \mbox{and\ so\ on\ recursively}
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(A-\lambda I) = $}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline497: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\mbox{det}\ \left\lbrack \begin{array}{llll}
a_{11} - \lambda & a_{12} \\ 
a_{21} & a_{22} - \lambda \\ 
& a_{32} \\ 
\\ 
\\ 
&&&a_{nn} - \lambda \end{array} \right\rbrack
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$( A - \lambda I ) = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline499: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
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}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\lambda^n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline505: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(-1)^n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline507: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\lambda = \lambda_1, \lambda_2, \ldots \lambda_n
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(A - \lambda I) = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline511: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bullet$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline513: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\lambda_k = 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline519: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}A v_k & = & \lambda_k v_k \\ 
\left. A_{ij} v_k \right)_j & = & \lambda_k \left. v_k \right)_i
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\left. v_k \right)_i^*$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline525: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$i=1\ldots n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline527: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\langle v_k \vert A \vert v_k \rangle & = & \lambda_k \langle v_k
\vert v_k \rangle \ \mbox{or} \\ 
\sum_{i,j=1}^n \left. v_k \right)_i^* A_{ij} \left. v_k \right)_j
& = & \lambda_k \sum_i \left. v_k \right)_i^* \left. v_k \right)_i
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mbox{LHS}\ = \lambda_k\, \mbox{RHS}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline529: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\mbox{RHS} & = & \sum_i \left\vert \left. v_k \right)_i \right\vert^2 \\ 
& = & \left\langle v_k \vert v_k \right\rangle 
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}\mbox{LHS} & = & \sum_{i,j=1}^n \left. v_k \right)_i^* A_{ij} \left.
v_k \right)_j\\ 
\mbox{LHS}^* & = & \sum_{i,j=1}^n \left. v_k \right)_i 
\underbrace{A_{ij}^*}_{\mbox{as\ hermitean}\ \ A^*_{i,j} =  A_{ji}}
\left. v_k \right)_j^* \\ 
& = & \sum_{i,j=1}^n \left. v_k \right)_i A_{ji} \left. v_k \right)_j^*
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 243
\mbox{LHS*} & = & \sum_{i,j=1}^n \left. v_k \right)_i^* A_{ij}
\left. v_k \right)_j =  \mbox{LHS} \\ 
&&\mbox{Therefore,\ LHS\ is\ real}
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\langle x | A | y \rangle^* & = & \langle y | A^+ | x \rangle \\ 
& = & \langle y | A | x \rangle \mbox{if}\ A\ \mbox{Hermitean}
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\langle x | = \langle y |$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline537: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\langle x | A | x \rangle^* = \langle x | A | x \rangle$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline539: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\lambda_k = \frac{\sum_{i,j=1}^n \left. v_k^* \right)_i A_{ij}
\left. v_k \right)_j}{\sum_{i=1}^n \left. v_k^* \right)_i \left. v_k 
\right)_i} \end{displaymath}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray*}% latex2html id marker 264
A | v_1 > & = & \lambda_1 | v_1 > \\ 
A | v_2 > & = & \lambda_2 | v_2 > \\ 
\langle v_2 | A | v_1 \rangle & = & \lambda_1 \langle v_2 | v_1
\rangle \hspace{1em} ( \alpha ) \\ 
\langle v_1 | A | v_2 \rangle & = & \lambda_2 \langle v_1 | v_2
\rangle
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\langle v_2 | A^+ | v_1 \rangle & = & \lambda_2^* \langle v_2 | v_1
\rangle \\ 
\langle v_2 | A | v_1 \rangle & = & \lambda_2 \langle v_2 | v_1
\rangle \hspace{1em} ( \beta ) \\ 
\mbox{as}\ A = A^+ && \lambda_2 = \lambda_2^*
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 545
$(\alpha )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline545: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(\beta )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline547: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\left( \lambda_1 - \lambda_2 \right) \langle v_2 | v_1 \rangle = 0
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\lambda_1 = \lambda_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline549: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\langle v_2 | v_1 \rangle
= 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline551: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vert v_1>$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline553: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vert v_2>$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline555: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\lambda_1 = \lambda_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline557: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vert v_2>$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline559: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vert v_2^\prime >$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline561: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vert v_2^\prime >$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline563: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vert v_1>$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline565: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vert v_2>$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline567: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$< v_1
\vert v_2^\prime > = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline569: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}\langle v_k | v_k \rangle  =  \sum_{i=1}^n \left. v_k^* \right)_i
v_k \left. \right)_i =  1
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\langle v_k | v_l \rangle = 0 \hspace{2em} l\neq k
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U^+$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline579: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}U^+ = \left\lbrack \begin{array}{cccccc}
\left. v_1 \right)_1 &&&&& \left. v_n \right)_1 \\ 
\left. v_1 \right)_2 \\ 
\left. v_1 \right)_3 \\ 
\\ 
\\ 
\left. v_1 \right)_n &&&&& \left. v_n \right)_n \end{array}
\right\rbrack
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U^+ U = I$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline583: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{A}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline589: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\tilde{A} = U^* \left\lbrack \begin{array}{cccc}
\lambda_1 \\ 
& \lambda_2 & 0 \\ 
0 \\ 
& & & \lambda_n \end{array} \right\rbrack\ U
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\uparrow$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline593: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\tilde{A} | v_k > = \lambda_k | v_k >
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{A}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline595: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}| x > = \sum_{k=1}^n x_k | v_k >
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$| v_k >$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline603: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\tilde{A} | x > & = & \sum_{k=1}^n x_k \tilde{A} | v_k > \\ 
& = & \sum_{k=1}^n x_k A | v_k > \\ 
& = & A | x > \\ 
\mbox{and\ we\ have\ proven}\ \tilde{A} & \equiv & A
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{displaymath}A = U^+\ \mbox{Diag.}\  \left\lbrack \begin{array}{ccc}
\lambda_1 \\ 
& \ddots \\ 
&& \lambda_n \end{array} \right\rbrack U
\end{displaymath}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A^+ = A$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline607: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A^+ = A^T$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline611: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A^T = A$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline613: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\underline{\rm vectors}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline615: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(A-\lambda_i I) | v_k >$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline617: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\lambda_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline619: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}U^+ & = & U^T \\ 
U^T U & = & I \\ 
U & \mbox{is} & \mbox{ORTHOGONAL}
\end{eqnarray*}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(n)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline623: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline625: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{d}{dx}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline627: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{d}{dx} f(x) = g(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline629: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{d}{dx}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline631: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}I = \int^b_a g(x) \frac{d}{dx} f(x) dx = \langle g\vert D\vert f\rangle
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{d}{dx}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline635: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}I  = & - & \int^b_a f(x)\, \frac{d}{dx}\, g(x)\, dx \\ 
     & + & \big\lbrack f(x) g(x) \big\rbrack^b_a \\ 
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\big\lbrack f(x) g(x) \big\rbrack^b_a  = f(b)\, g(b) - f(a)\, g(a)
\end{displaymath}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$i\frac{d}{dx}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline637: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}f(b) & = & f(a) \\ 
g(b) & = & g(a) \\ 
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\int^b_a g(x) \frac{d}{dx} f(x)\, dx = - \int^b_a f(x) \frac{d}{dx} g(x)\, dx
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$L = i \frac{d}{dx}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline639: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\underline{\rm
boundary\ conditions}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline641: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f(a) = f(b)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline643: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\nabla^2 = \frac{d^2}{dx^2} = -L\,L$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline649: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\left(\nabla^2 \right)^\dagger = -L^\dagger L^\dagger = -L\,L = 
\nabla^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline651: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(A\,B)^\dagger = B^\dagger A^\dagger$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline653: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\nabla^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline655: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\frac{d^2}{dx^2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline657: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}

\end{document}
