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\newtheorem{activity}{Sample Application }[section]
\newtheorem{project}{Project}[section]
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\newtheorem*{weak_duality}{The Weak Duality Theorem}
\newtheorem*{strong_duality}{The Strong Duality Theorem}
\newtheorem*{duality_theorem}{The Duality Theorem}
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\newcommand{\LPC}{\emph{FuelPro Petroleum Company }}
\newcommand{\LP}{\emph{FuelPro }}

\newcommand{\CMC}{\emph{ConPro Manufacturing Company }}
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%LP
%\begin{align}
% \mbox{maximize } z &=10x_1+8x_2  \\
%	\mbox{ subject to} \notag\\
%	x_1 \phantom{ +20x_2} &\leq 10 \notag\\
%	x_1+2x_2 &\leq 20  \notag\\
%	10x_1+2x_2 &\leq 120 \notag \\
%	x_1,x_2 &\geq 0.\notag
%	\end{align}	

%TABLEAU

%\begin{table}[h]
%\centering
%\begin{tabular}{|c||c|c|c|c|c||c|}
%\hline
%$z$ & $x_1$ & $x_2$ & $s_1$ & $s_2$ & $s_3$ & RHS \\
%\hline \hline
%1 & 0 & -8 & 10 & 0 & 0 & 100 \\ \hline
%0 &  1 & 0 & 1 & 0 & 0 & 10 \\ \hline
%0 &  0 & 2 & -1 & 1 & 0 & 10 \\ \hline
%0 &  0 & 7 & -10 & 0 & 1 & 20 \\ \hline
%\end{tabular}
%\caption{Tableau after first iteration}\label{Ta:petroleum_second_tableau}
%\end{table}

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%Section: \label{S:Name}
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%\hfil {  \includegraphics{c:/images/OR/blending_feasible_region.eps} }  \hfil
%\caption{Feasible region for Fishback Petroleum LP}\label{Fi:blending_feasible_region}
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%\caption{Quadratic form $\ds f(x_1,x_2)=x_1^2+4x_1x_2+x_2^2$}
%\label{Fi:optimality_sufficient_SOC_quad_form_INDEF}
%\end{figure}

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%	$\det(3B)$ &   $\displaystyle \det \left( \begin{bmatrix} 1 & 2 & 4 \\ 2 & 0 & -2 \\ 1 & 2 & 2 \end{bmatrix} \right)$
%	\end{tabular}

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%	\raisebox{.75in}{\begin{tabular}[b]{c|c} $x$ & $y$ \\ \hline a & b \\ c & d \\ e & f  \end{tabular} } &	
%	\hspace{1in} \includegraphics[2in,2in]{c:/mth310/project/Hamilton.bmp}
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%\caption{{\scriptsize Table and graph for $f$.}}
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%	\end{cases}

%GENERAL NLP
%\begin{equation}\label{E:diff_Gradient_general_NLP}
%\mbox{minimize (or maximize) } f(\vecx) \mbox{ where }\vecx \in S.
%\end{equation}

%GENERAL LP
%\begin{align}
% \mbox{minimize } w  \\
%	\mbox{ subject to} \notag\\
%	\vecy A&\leq w\vece^t \notag \\
%	\vecy\cdot\vece &=1 \notag  \\
%	\vecy &\geq \zerovec. \label{E:Duality_Steves_strategy}
%\end{align}

%WEB URL
%\texttt{http://www.phy.syr.edu/courses/java-suite/crosspro.html}

%MAPLE FILE NAME
%\textbf{extrema.mws}

% POSITIONING
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% WITHIN ANOTHER ENVIRONMENT \centering, \raggedright, \raggedleft

%HEADING AND DIRECTIONS
\begin{document}

\noindent
Here is a list of Maple commands used at various points in this text, organized by package name. Each command is accompanied by a brief explanation of its syntax as well as an example. To learn more about a particular command, type \mf{?command name} at the command prompt. To load a particular package, type \mf{with(\emph{package name})}.


%*************************************************************
%*******************Functions****************************
%*************************************************************


\paragraph{Functions}

\be

\item \mf{f:=x-> \emph{expression in x}:} 

\noi Constructs a function, named \mf{\emph{f}}
whose rule is governed by the given \mf{expression}.

    \newpar
    \emph{Example:}
    \begin{maplelist}
    \item \mf{f:=x->x $\wedge$ 2 -x;} 
    
    \noi Produces the function $\ds f(x)=x^2-x.$
    \end{maplelist}

Functions of more than one variable are similarly defined.

    \newpar
    \emph{Example:}
    \begin{maplelist}
    \item \mf{f:=(x1,x2)->x1 $\wedge$ 2 -x2 $\wedge$ 2;} 
    
    \noi Produces the function $\ds f(x_1,x_2)=x_1^2-x_2^2$.
    \end{maplelist}

 The \mf{unapply} command defines functions as well.

      \newpar
    \emph{Example:}
    \begin{maplelist}
    \item \mf{f:=unapply(x1 $\wedge$ 2 -x2 $\wedge$ 2,[x1,x2]);} 
    
    \noi Produces the function $\ds f(x_1,x_2)=x_1^2-x_2^2$.
    \end{maplelist}

 \ee
 
 %*************************************************************
 %*******************Plot Structures****************************
 %*************************************************************

 \paragraph{Plot Structures}
 \be

%*******************Plot and Plot3d****************************

 \item \mf{plot(expression,options);}
 
 \noi Plots an \mf{expression} of a single variable using specified \mf{options}, which dictate
 plot interval, graph color, etc.

 \newpar
    \emph{Examples:}
    \begin{maplelist}
    \item \mf{f:=x->x$\wedge$2}
 \item \mf{plot(f(x),x=-1..3,color=red);} 
 
 \noi Plots the function $f(x)=x^2$ in red
 on the interval $[-1,3]$
 \end{maplelist}
 
 \newpar
 For a complete list of \mf{plot} options, type \mf{?plot[options]} at the command prompt.

\item \mf{plot3d(expression,options);}
 
 \noi Plots an \mf{expression} in two variables using specified \mf{options}, which dictate
 surface plot style, plot region, color, etc.

 \newpar
    \emph{Example:}
    \begin{maplelist}
    \item \mf{f:=(x1,x2)->x1*x2}
 \item \mf{plot3d(f(x1,x2),x1=0..2,x2=-2..2,style=wireframe,color=blue);} 
 
 \noi Plots the function $f(x_1,x_2)=x_1x_2$ in blue, wireframe style on the region $0 \leq x_1 \leq 2$
     and $-2 \leq x_1 \leq 2$.
 \end{maplelist}

\newpar
 For a complete list of \mf{plot3d} options, type \mf{?plot3d[options]} at the command prompt.
 
 \item \mf{plot3d(expression,options);}
 
 \noi Plots an \mf{expression} in two variables using specified \mf{options}, which dictate
 surface plot style, plot region, color, etc.

 \newpar
    \emph{Example:}
    \begin{maplelist}
    \item \mf{f:=(x1,x2)->x1*x2}
 \item \mf{plot3d(f(x1,x2),x1=0..2,x2=-2..2,style=wireframe,color=blue);} 
 
 \noi Plots the function $f(x_1,x_2)=x_1x_2$ in blue, wireframe style on the region $0 \leq x_1 \leq 2$
     and $-2 \leq x_2 \leq 2$.
 \end{maplelist}
 
 %*******************Implicitplot****************************
 
 \item \mf{implicitplot(relation,options);}
 
 \noi Plots a \mf{relation} in two variables using specified \mf{options}, which dictate
 surface plot region, color, etc.

 \newpar
    \emph{Example:}
    \begin{maplelist}
 \item \mf{implicitplot(x1$\wedge$-x2$\wedge$=1,x1=-5..5,x2=-5..5,color=green,thickness=3);}
     
     \noi Plots the relation $x_1^2-x_2^2=1$ in green, thickness level 3 on the region $-5 \leq x_1 \leq 5$
     and $-5 \leq x_2 \leq 5$.
 \end{maplelist}
 
 %*******************ContourPlot****************************

 \item \mf{contourplot(relation,options);}
 
 \noi Creates a contour plot of an \mf{expression} in two variables using specified \mf{options}, which dictate plot region, color, contour values, etc.

 \newpar
    \emph{Example:}
    \begin{maplelist}
 \item \mf{contourplot(x1$\wedge$-x2$\wedge$=1,x1=-5..5,x2=-5..5,color=blue,contours=[-2,-1,0,1,2]);} 
     \noi Creates blue contour plot on the region $-5 \leq x_1 \leq 5$
     and $-5 \leq x_2 \leq 5$ of the expression $z=x_1^2-x_2^2=1$ using $z$ values of $z=-2,-1,0,1,2$.
 \end{maplelist}
 
 %*******************Inequal****************************

 \item \mf{inequal(inequalities,options);}
 
 \noi Plots the region in the plane consisting of points
 that satisfy the given inequalities. The inequalities must be linear, allow for
 equality, 
 and should be enclosed within brackets and separated by commas. The options
 dictate the plot region, the color of the region satisfying the inequalities, (\mf{optionsfeasible}), the color of the boundary of the region, (\mf{optionsclosed}), 
 and the color of the points not satisfying at least one inequality, (\mf{optionsexcluded}). 
 This command is contained in the \mf{plots} package.
 
 \newpar
    \emph{Example:}
    \begin{maplelist}
    \item \mf{with(plots):}
 \item \mf{inequal([x1+x2 <= 3, x2 >= x1], x1 = 0 .. 5, x2 = 0 .. 5, optionsfeasible = (color = red), optionsclosed = (color = green), optionsexcluded = (color = yellow));} 
     
     \noi Plots in red the set of points, $(x_1,x_2)$, that satisfy the 
     inequalities, $x_1+x_2 \leq 3$ and $x_2\geq x_1$ and that belong to the plot region
     $0 \leq x_1 \leq 5$
     and $0 \leq x_2 \leq 5$. The boundary of the region is colored green, and the points
     not satisfying at least one inequality are colored yellow.
 \end{maplelist}
 
 \newpar
This command also has coloring options for situations when at least one inequality is strict, i.e., involves $<$ or $>$ as opposed to $\leq$ or $\geq$.

 
 %*******************Pointplot****************************

  \item \mf{pointplot(list of points,options);}
 
 \noi Plots the specified \mf{list of points} using specified \mf{options}. Each point should consist of two numbers,
 separated by commas and enclosed within brackets. Then these points themselves
 are separated by commas and enclosed within brackets again. Options permit specifying
 the point color, the point symbol, (asterisk, box, circle, cross, diagonalcross, diamond, point, solidbox, solidcircle, soliddiamond), and the size of this symbol, whose default
 value is 15. This command is contained in the \mf{plots} package.

 \newpar
    \emph{Example:}
    \begin{maplelist}
    \item \mf{with(plots):}
 \item \mf{pointplot([[-1,3],[0,0],[3,7]],symbol=box,symbolsize=20);}
     
     \noi Plots, as red boxes, the ordered pairs, $(-1,3)$, $(0,0)$, and $(3,7)$.
     The size of the symbol is slightly larger than the default value.
 \end{maplelist}

 \newpar
A similar command, \mf{pointplot3d}, exists for pointing points in $\R^3$.
 
 %*******************Display****************************

 \item \mf{display(plot structures,options);}
 
 \noi Superimposes on a single set of axes, the given \mf{plot structures}, all of which belong to $\R^2$ or all of which belong to $\R^3$, using the specified
 \mf{options}. The plot structures themselves must be separated by commas and
 enclosed within brackets or braces.
 
 \newpar
 This command, which is located in the \mf{plots} package, is most frequently to superimpose different plot structures that have been previously created and assigned different names.

 \newpar
    \emph{Example:}
   \begin{maplelist}
    \item \mf{with(plots):}
 \item \mf{graph1:=pointplot([[-1,3],[0,0],[3,7]],symbol=box,symbolsize=20):}
 \item \mf{graph2:=inequal([x1+x2 <= 3, x2 >= x1], x1 = 0 .. 5, x2 = 0 .. 5, optionsfeasible = (color = red), optionsclosed = (color = green), optionsexcluded = (color = yellow)):}
 \item \mf{display([graph1,graph2],axes=framed);} 
 
 \noi Displays the superposition of the set of points specified by \mf{graph1}, along with the region specified by \mf{graph2}
 \end{maplelist}
 


 \ee 

%*************************************************************
 %*******************Conditional Statements and Loops****************************
 %*************************************************************

 \paragraph{Conditional Statements and Looping}

 \be

 \item \mf{if condition then operation end if:}
 
 \noi Performs given \mf{operation} provided
 \mf{condition} is true.

 \newpar 
    \emph{Example:}
    \begin{maplelist}
    \item \mf{a:=5:}
 \item \mf{if a > 2 then x:=5 end if;} 
 
 \noi Assigns the value of 5 to $x$ since $a>2$.
 \end{maplelist}

 \item \mf{for i from m to n by j do task end do:} 
 
 \noi Perform \mf{\emph{task}}
 using index value \mf{\emph{i}}, which varies from \mf{\emph{m}} to \mf{\emph{n}} in increments of \mf{\emph{j}}.

 \newpar
    \emph{Example:}
    \begin{maplelist}
    \item \mf{for i from 0 to 20 by 4 do print(i) end do;} 
    
    \noi Prints the values of 0, 4, 8, 12, 16, and 20.
    \end{maplelist}


\ee

%*************************************************************
 %*******************Arrays and Lists****************************
 %*************************************************************
 
 \paragraph{Arrays and Lists}
 \be
 
 \item \mf{array(i..j,m..n);} 
 
 \noi Creates an array in two indices, with the first integer index varying from \mf{i}
 to \mf{j} and the second from \mf{m} to \mf{n}. 
 
  \newpar
    \emph{Example:}
    \begin{maplelist}
    \item \mf{A:=array(0..2,1..3);}

 \noi Creates an array, \mf{A}, consisting of 9 entries, where the indices
 vary from 0 to 2 and 1 to 3, respectively.
 \end{maplelist}
 
 \newpar
 Array entries are frequently assigned values using looping structures.
 
  \newpar
    \emph{Example:}
    \begin{maplelist}
    \item \mf{A:=array(0..2,1..3);}
    \item \mf{for i from 0 to 2 do for j from 1 to 3 do A[i,j]:=i+j:od:od:}

 \noi Creates an array, \mf{A}, consisting of 9 entries, where the indices
 vary from 0 to 2 and 1 to 3, respectively. Each entry is the sum of the two 
 corresponding indices.
 \end{maplelist}
 
 \item \mf{[entries];}

 \noi Creates an ordered list using \mf{entries}. The values of \mf{entries} must be separated by commas. Values in the list can be extracted using subscripts; the commands \mf{nops} determines the size of the list, and \mf{op} removes the outer brackets.
 
  \newpar
    \emph{Example:}
    \begin{maplelist}
    \item \mf{L:=[1,2,4,8,16,32]}
    \item \mf{L[3]}
    
    \noi Returns 4.
    
    \item \mf{nops(L);}
    
    \noi Returns 6.
    
    \item \mf{op(L);}
    
    \noi Returns $1,2,4,8,16,32$.
 \end{maplelist}
 
 
 
 \ee
 
%*************************************************************
 %*******************Sequences and Sums****************************
 %*************************************************************

 \paragraph{Sequences and Sums}
 \be

 \item \mf{seq(expression,i=m..n);}
 
 \noi Creates a sequence of values that results when the integer index value \mf{i},
  which varies between \mf{m} and \mf{n}, is substituted
 into \mf{expression}. 

\newpar
    \emph{Example:}
    \begin{maplelist}
    \item \mf{seq(3$\wedge$i,i=-2..3);}

    \noi Returns $\ds \frac{1}{9}, \frac{1}{3}, 1, 3, 9, 27$.
    
 \end{maplelist}

\item \mf{sum(expression,i=m..n);}

\noi Computes a closed form for the sum that results when the integer index
value \mf{i} is substituted into \mf{expression} and the resulting sequence of
values are added together.

\newpar
    \emph{Example:}
    \begin{maplelist}
    \item \mf{sum((1/2)$\wedge$i,i=0..infinity);}

    \noi Returns the geometric series sum of 2.
    \end{maplelist}

\item \mf{add(expression,i=m..n);}

\noi Computes the explicit sum that results when the integer index
value \mf{i} is substituted into \mf{expression} and the resulting sequence of
values are added together.

\newpar
    \emph{Example:}
    \begin{maplelist}
    \item \mf{sum(2$\wedge$i,i=-2..6);}

    \noi Returns $\ds \frac{511}{4}$.
    \end{maplelist}



 \ee
 

%*************************************************************
%*******************Linear Algebra ****************************
%*********************
\paragraph{Linear Algebra}

\be

%*******************Vectors****************************
\item \mf{Vector(entries)}: 

\noi Constructs a vector having components given by \mf{entries}. The entries should be separated by commas and enclosed with brackets. By default, Maple assumes a vector is a column vector. To enter a row vector instead, type \mf{Vector[row](entries)}.
    
    \newpar
 \emph{Examples:}
    \begin{maplelist}
    \item \mf{Vector([1,2]);} 
    
    \noi Produces the column vector,
    $\ds \begin{bmatrix} 1 \\ 2 \end{bmatrix}$.
    \item \mf{Vector[row]([1,2]):} 
    
    \noi Produces the row vector,
    $\ds \begin{bmatrix} 1 & 2 \end{bmatrix}$.
    \end{maplelist}
    
    \noi 
    Alternatively, a vector can be entered using $<$, $>$, notation, e.g., \mf{<1,2>}.

 %************************Matrices

\item \mf{Matrix(m,n,entries);} 

\noi Constructs an \mf{m} by \mf{n} matrix having entries, read
across rows given by \mf{entries}. The entries should be separated by commas and enclosed with brackets.

\newpar
 \emph{Example:}
  \begin{maplelist}
  \item \mf{Matrix(2,3,[1,4,5,-3,2,5]);} 
  
  \noi Produces the matrix,
    $\ds \begin{bmatrix} 1 & 4 & 5 \\ -3 & 2 & 5 \end{bmatrix}$.
   \end{maplelist}


\noi Alternatively, a matrix can be entered by combining vectors and using the $<$, $>$, notation. Vectors are augmented using $|$ and stacked via the comma.

\noi \emph{Example:}
  \begin{maplelist}
  \item \mf{v1:=<2,3>:}
  \item \mf{v2:=<-4,0>:}
  \item \mf{<v1|v2>;} 
  
  \noi Produces the matrix, $\ds \begin{bmatrix} 2 & -4 \\ 3 & 0 \end{bmatrix}$.
  \item \mf{<v1,v2>;} 
  
  \noi Produces the matrix, $\ds \begin{bmatrix} 2 \\ 3 \\ -4 \\ 0 \end{bmatrix}$.
    \end{maplelist}

%************************Zero Matrices and Vectors

\item \mf{ZeroMatrix(m,n);} 

\noi Constructs an \mf{m} by \mf{n} zero matrix;
\mf{ZeroVector(m);} constructs a column vector with \mf{m} entries;

\newpar
 \emph{Example:}
  \begin{maplelist}
  \item \mf{ZeroMatrix(2,3);} 
  
  \noi Produces the matrix,
    $\ds \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$.
   \end{maplelist}


%************************Identity Matrices

\item \mf{IdentityMatrix(m);} 

\noi Constructs an \mf{m} by \mf{m} identity matrix.

\newpar
 \emph{Example:}
  \begin{maplelist}
  \item \mf{IdentityMatrix(2);} 
  
  \noi Produces the matrix,
    $\ds \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.
   \end{maplelist}

%************************Unit Vector

\item \mf{UnitVector(m,n);} 

\noi Constructs a column vector in $\R^n$ having an entry
of 1 in its \mf{m} component and zero in all others.

\newpar
 \emph{Example:}
  \begin{maplelist}
  \item \mf{UnitVector(2,4);} 
  
  \noi Produces the vector,
    $\ds \begin{bmatrix} 0 \\ 1 \\ 0 \\0 \end{bmatrix}$.
   \end{maplelist}

%************************Diagonal Matrix

\item \mf{DiagonalMatrix(entries);} 

\noi Constructs a square, diagonal matrix having diagonal entries given by \mf{entries}. The entries should be separated by commas and enclosed with brackets.

\newpar
 \emph{Example:}
  \begin{maplelist}
  \item \mf{DiagonalMatrix([1,3,-5]);} 
  
  \noi Produces the matrix,
    $\ds \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -5 \end{bmatrix}$.
   \end{maplelist}

%************************Extract rows and columns

\item \mf{Row(M,m);} 

\noi Extracts, as a row vector, row \mf{m} of the matrix
    \mf{M}. Likewise, \mf{Column(M,n)} extracts, as a column vector, column \mf{n} of the matrix \mf{M}.

\newpar
 \emph{Example:}
 \begin{maplelist}
  \item \mf{A:=Matrix(2,3,[1,4,5,-3,2,5]):}
 \item \mf{Row(A,2);} 
 
 \noi Produces the vector, $\ds \begin{bmatrix} -3 & 2 & 5\end{bmatrix}$.
    \end{maplelist}

 %************************Row Operations

\item \mf{RowOperation(M,operation);} 

\noi Performs a specified row operation,
\mf{operation} on the
matrix \mf{M}. There are three such operations.

\begin{maplelist}
\item \mf{RowOperation(M,j,k);} 

\noi Multiplies row
\mf{j} of \mf{M} by \mf{k}.
\item \mf{RowOperation(M,[i,j]);} 

\noi Interchanges rows \mf{i}
 and \mf{j} of \mf{M}.
\item \mf{RowOperation(M,[i,j],k);} 

\noi Adds  \mf{k}
times row \mf{j} to \mf{i}.
\end{maplelist}

\newpar
\emph{Example:} 
 \begin{maplelist}
  \item \mf{A:=IdentityMatrix(3):}
 \item \mf{RowOperation(A,3,4);} 
 
 \noi Produces the matrix,
 $\ds \begin{bmatrix} 1 & 0 & 0 \\
 0 & 1 & 0 \\
 0 & 0 & 4
 \end{bmatrix}$.

 \item \mf{RowOperation(A,[1,3]);} 
 
 \noi Produces the matrix,
 $\ds \begin{bmatrix} 0 & 0 & 1 \\
 0 & 1 & 0 \\
 1 & 0 & 0
 \end{bmatrix}$.

 \item \mf{RowOperation(A,[1,3],-2);} 
 
 \noi Produces the matrix,
 $\ds \begin{bmatrix} 1 & 0 & -2 \\
 0 & 1 & 0 \\
 0 & 0 & 1
 \end{bmatrix}$.

\end{maplelist}
Note: The option, \mf{inplace=true}, replaces the value of the matrix with that obtained
by performing the specified row operation. For example, \mf{RowOperation(M,[1,2], inplace=true)} interchanges rows 1 and 2 of \mf{M} and overwrites the value of \mf{M} with
this new matrix.

%************************Pivot

\item \mf{Pivot(M,i,j);} 

\noi Pivots on the entry in row \mf{i}, column
 \mf{j} of the matrix
    \mf{M}.

\newpar
 \emph{Example:}
 \begin{maplelist}
  \item \mf{A:=Matrix(3, 3, [1, 4, 5, -3, 2, 5, 7, 0, -1]):}
 \item \mf{Pivot(M,1,3);} 
 
 \noi Produces the matrix,
 $\ds \begin{bmatrix} 1 & 4 & 5 \\ -4 & -2 & 0 \\ \frac{36}{5} & \frac{4}{5} & 0 \end{bmatrix}$.
    \end{maplelist}

%************************Row and Column Dimension

\item \mf{RowDimension(M);} 

\noi Determines the row dimension of the matrix
    \mf{M}. Likewise, \mf{ColumnDimension(M);} determines the column
    dimension of \mf{M}.

\newpar 
 \emph{Example:}
 \begin{maplelist}
  \item \mf{A:=Matrix(2,3,[1,4,5,-3,2,5]):}
 \item \mf{RowDimension(A)} 
 
 \noi Yields 2.
 \item \mf{ColumnDimension(A)} 
 
 \noi Yields 3.
    \end{maplelist}

%************************RREF and REF

\item \mf{GaussianElimination(M);} 

\noi Performs Gaussian elimination
on the matrix  \mf{M} and produces the row echelon form.

\item \mf{ReducedRowEchelonForm(M);} 

\noi Computes the reduced row
echelon form of \mf{M} .

%************************Determinant

\item \mf{Determinant(M);} 

\noi Computes
the determinant of the matrix \mf{M}.

%************************Inverse

\item \mf{MatrixInverse(M);} 

\noi Calculates
the inverse of the matrix \mf{M}. If the matrix is not invertible, an error message
is returned.

%************************Eigenvalues

\item \mf{Eigenvalues(M);}: 

\noi Calculates
the eigenvalues of the matrix \mf{M}.

%************************Eigenvectors

\item \mf{Eigenvectors(M);} 

\noi Calculates
the eigenvectors and corresponding eigenvalues of the matrix \mf{M}. The eigenvalues
are returned as a column vector, and the eigenvectors as corresponding
columns in a matrix.

\newpar
 \emph{Example:}
 \begin{maplelist}
  \item \mf{A:=Matrix(3,3,[1, 0, 0, 2, 3, 0, 4, 5, 3]):}
 \item \mf{Eigenvectors(A)} 
 
 \noi Yields $\ds \begin{bmatrix} 1 \\ 3 \\ 3 \end{bmatrix}$,
 along with $\ds \begin{bmatrix} 2 & 0 & 0 \\ -2 & 0 & 0 \\ 1 & 1 & 0 \end{bmatrix}$.
 Thus, the matrix \mf{A} has eigenvectors
 $\ds \begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix}$ (eigenvalue of 1) and
 $\ds \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$ (repeated eigenvalue of 3).
\end{maplelist}

 %************************IsDefinite

 \item \mf{IsDefinite(M,'query'=q);} 
 
 \noi Tests whether the matrix \mf{M} is
 positive (semi-) definite, negative (semi-) definite, or indefinite. The returned
 value is \emph{true} or \emph{false}, and the parameter
 \mf{\emph{q}} specifies which matrix form is to be determined. Choices
 for \mf{\emph{q}} consist of \mf{'positive\_definite'},
 \mf{'positive\_semidefinite'},
 \mf{'negative\_definite'}, \mf{'negative\_semidefinite'}, and \mf{'indefinite'}.

 \noindent
 If the matrix has symbolic
 entries, conditions on these entries required for \mf{q} to be
 satisfied are returned.

\newpar
 \emph{Examples:}
 \begin{maplelist}
  \item \mf{A:=Matrix(3,3,[1, 0, 0, 2, 3, 0, 4, 5, 3]):}
 \item \mf{IsDefinite(A,'query'='positive\_definite')} 
 
 \noi Returns \emph{true}.
  \item \mf{A:=Matrix(3,3,[1, 0, 0, 2, 3, 0, 4, 5, x]):}
 \item \mf{IsDefinite(A,'query'='positive\_semidefinite');} 
 
 \noi Returns
 $\ds \Re(x) \geq 0$.
 \end{maplelist}

\ee

%*************************************************************
%*******************Linear Algebra ****************************
%*********************
\paragraph{\mf{Optimization}}



%*******************Vectors****************************

\end{document} 