% FILE anton1011.tex


%AR Chapter 10. Complex vector spaces
\section{Complex Vector Spaces}


Maple's functions work the same when entries are complex as when entries
are real. 
This chapter of \cite{AR} treats matrices with complex entries, e.g.
eigenvals - which you might apply on 
Example 5, [AR] \S10.6, where   % Works in both ed.s
%AR5 S9.6 Example 34, where
\begin{verbatim}
A:=array([[2,1+I],[1-I,3]]);
\end{verbatim}

Here are a few Maple functions relevant to vectors and matrices
with complex entries.
\index{innerprod}\index{dotprod}\index{htranspose}
\begin{maple}
# Maple V has a function innerprod which is more general than the
# dotprod function, though most useful with real vectors and matrices.
with(linalg,dotprod,innerprod):
u:=array([I,1+I]): v:=array([-2*I,1+I]):
simplify(dotprod(u,v));
                                     0
innerprod(u,v);
                                  2 + 2 I
simplify(dotprod(u,v,'orthogonal'));
                                  2 + 2 I
#########################################################################
# Example 2 Section10.6 - AR6 p517; AR7 p560
# Show that the matrix A defined below is unitary.
# Maple 5.0.1 does not have a unitary test like orthog
# orthog(A) is false as it should be.
# htranspose and be said to people, not Maple, 'hermitian_transpose'.
with(linalg,htranspose,multiply):
A:=array([[(1+I)/2,(1+I)/2],[(1-I)/2,(-1+I)/2]]):
multiply(htranspose(A),A);
                                  [ 1  0 ]
                                  [ 0  1 ]
# This verifies that A is unitary.
##########################################################################
\end{maple}

\bigskip

\subsection*{Supplement. Jordan forms ([J] \S3.2.2)}

Jordan forms are not treated in \cite{AR}.
See more advanced linear algebra books,
or the very brief account in \cite{J}.
Maple V has a
{\tt jordan}\index{jordan}
function.
%and examples are on the disk supplement.

\vfill
\eject

% Chapter 11. 
\section{Applications of Linear Algebra}


\subsection*{Contents of this chapter from [AR]}
The material from this chapter is treated in Maple Worksheets
rather than in this document.
How much is set up in student labs will depend on your lecturer.

\par\noindent
\S AR6.1. (\cite{J} \S1.6.)
Constructing curves and surfaces through given points.
Uses {\tt det}.

\par\noindent
\S AR6.2. Plane geometry.
(There are also alternate approaches to some of these problems 
using the {\tt geometry} package from Maple V 
given on Worksheet.)

\par\noindent
\S AR6.3. Equilibrium of rigid bodies: see also \cite{MK}
J.L. Meriam and L.G. Kraige,
{\it Engineering mechanics, Vol 1, Statics - SI edition.}
(Wiley).
Uses  {\tt linsolve}.
(See example given on Worksheet.)

\par\noindent
\S AR6.4. (\cite{J} \S1.6.) Graph theory. 
            (Some examples are given on Worksheet. 
There are other uses, e.g. 
           in connection with connectivity information 
in truss statics problems - [AR] \S AR6.3, 
           in computer graphics -  [AR]  \S AR6.10, 
p.d.e. discretisations -  [AR] \S AR6.11, 
           in networks -  [AR] \S AR6.19.)

\par\noindent
\S AR6.5. Theory of games. 
            (The approach given on Worksheet, 
uses Maple's {\tt simplex} package.)

\par\noindent
\S AR6.6. The assignment problem. 
(Again, this is best solved using  Maple's {\tt simplex} package.)

\par\noindent
\S AR6.7. Markov chains. 

\par\noindent
\S AR6.8. (\cite{J} \S1.6.) Leontief economic models. 

\par\noindent
\S AR6.9. Forest management

\par\noindent
\S AR6.12. Some applications in genetics

\par\noindent
\S AR6.13. Age specific population growth

\par\noindent
\S AR6.14. Harvesting of animal populations
(The approach given on Worksheet, 
uses the {\tt linalg} {\tt eigenvals} function.)

\par\noindent
\S AR6.10. Computer graphics (See example given on Worksheet.)

\par\noindent
\S AR6.11. (Not yet treated.)

\par\noindent
\S AR6.15. Least squares. 
(See elsewhere in this document and on Worksheet.)

\par\noindent
\S AR6.16. (Cubic splines. See {\tt spline} in Maple Vr2.
Not yet treated.)

\par\noindent
\S AR6.17. Cryptography. (\cite{J} \S1.6.)
(See example given on Worksheet.)

\par\noindent
\S AR6.18. (Tomography. Not yet treated.)

\par\noindent
\S AR6.19. Electrical networks. Uses {\tt linsolve}.

\par\noindent
\S AR6.20. (Fractals. Not yet treated.)


\medskip
\subsection*{Calculus applications of {\tt linalg}}

We now advertise some other items in Maple's
{\tt linalg} package.


\subsubsection*{Multivariable differential calculus}

The derivative of a function $f$ from 
$\DoubleR^m$ to $\DoubleR^n$ is the 
$m$ by $n$ matrix of partial derivatives, and it 
is sometimes called the jacobian. 
If the names specifying the independent variables are in 
{\tt xlist}, the Maple function to find this matrix, 
in the {\tt linalg} package, is\index{jacobian}
\begin{verbatim}
jacf:= jacobian(f, xlist);
\end{verbatim}
See \cite{KN} for its uses.

The Maple functions 
{\tt grad}\index{grad} and 
{\tt hessian}\index{hessian}
occur in connection with finding and classifying 
critical points on surfaces.
See \cite{CH,KN}.



\subsubsection*{Vector analysis}

The following functions from {\tt linalg} 
have meanings appropriate to their names. 
(Use Maple 
online help if necessary).
\begin{verbatim}
grad(V, xlist)
diverge(fvec, xlist)
laplacian(V, xlist)
\end{verbatim}
For more on this, see \cite{CH,KN}.
%Cheung and Harer {\it Multivariate calculus with Maple V},
%(Wiley: 1993).

\bigskip

\section*{Acknowledgements by G. Keady}

I thank my undergraduate classes for the fact that
amongst them there were, I think, enough who shared 
my opinion that Maple was interesting and potentially useful, and 
my opinion that pushing buttons on a machine to make manipulative 
maths happen could be fun.
Student enthusiasm and discovery - 
mostly maths but sometimes items from CA
which are `correct but confusing'
such as the exercise on 3 by 3 real symmetric
0-1 matrices on page~\pageref{student} - has been vital to my 
maintaining my belief that Maple has a role in teaching.
The adaptation of curricula to the new CA tools which
allow students a more experimental approach to maths - guided by
staff - to help the experiments to be educationally useful -
is going to take time.

I thank Rod James of University of New South Wales for
the compliment paid in adapting -with my permission- my notes 
to the different (more advanced) text
in use at UNSW, and improving the notes. 
In particular, 
Rod sent me the code in the section on $QR$-factorisation,
in Chapter \ChIPS.
It is expected that some more of Rod's QR factorisation 
Maple code may appear in associated Maple Worksheets.
See page~\pageref{householder}.

I thank Howard Anton for his support.
I thank, especially, staff from Wiley USA
- Barbara Holland and Randy Jensen - for their trust in me
and my proposal that by posting to the Share Library and seeking
co-authors a better document will be built.
Examples such as \cite{BEJ,CH,EJ,H,KN}
%Kent Harris, Cheung and Harer, Evans and Johnson, 
%Kreyszig and Normington
supplied by Wiley are to act as models of
the quality which I and any future co-authors will strive to
achieve. 
I thank Bill Bauldry for early advice concerning the book \cite{BEJ},
for his interest in my efforts,
and for his article \cite{BH} showing that student discovery
as mentioned in the first paragraph of the acknowledgements,
see page \pageref{student},
using Maple has some similarities around the world.
The Wiley Australia staff, and particularly
Maureen Robbins, have been very cooperative and patient.

Last, but not least, thanks are due to Maple Waterloo Software
(i) for providing us with the program which has led to this effort,
and (ii) for the ftp Share Library which is such an appropriate method
for distribution of materials -such as these- which need frequent updating
as the software and texts (and supplements to texts)
move through different versions and editions.
\bigskip
%\vfill
%\eject
\endinput
