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\vspace*{3mm}
{\LARGE \bf{ \begin{center}
\underbar{ The ILAS Education Committee }\\[3mm]
\end{center} }}
{\Large \bf{\begin{center}
Report\\
on\\
Graduate Linear Algebra Courses
\end{center} } }
{\Large \begin{center}
March/November 1993\\[8mm]
\end{center} }
 
 
\noindent Based on our survey of US graduate course offerings in Linear
Algebra last fall, we have drawn up three separate course outlines for
graduate year--long courses in
\begin{center}
Theoretical Linear Algebra,\\
Numerical Linear Algebra, and\\
Applied Linear Algebra.
\end{center}
 
\noindent These outlines contain a large number of significant
modern topics in our field combined with basic and more
classical results. Several of the chosen topics occur in more than one
course. Take the $LU$ decomposition for example: a theoretical treatment
of the $LU$ decomposition would maybe emphasize the lower triangular -- upper
triangular group factorization of $GL_n$ and could possibly give a short
introduction into Della--Dora type group factorizations in a theoretical
vein. The numerical course would instead emphasize explicit index
manipulations and computations, as well as the operations count for $LU$,
while the applied
course might use it to establish rank, solvability and redundancy
criteria for linear systems of equations and integrate the $LU$
factorization thus with its applications.\\
 
\noindent We have chosen our outlined topics and their
interconnectedness in order to reflect the importance of our subject to modern
abstract and applied mathematics and to make graduate courses based on our
syllabi highly valuable to any modern mathematics, science or engineering
graduate program.\\
 
\noindent We shall refrain from specific textbook recommendations, but want to
assert that there has recently been a renaissance of excellent texts in
our field, well suited to achieve a higher level of awareness and
usefulness for Linear Algebra throughout the scientific community.\\
 
\begin{center}
Chair : Frank Uhlig  ( uhligfd@mail.auburn.edu )\\
David Carlson, Biswa N. Datta, Charlie Johnson, Steve Leon, Miki Neumann\\
Hans Schneider, President\\
The \, INTERNATIONAL\,  LINEAR\,  ALGEBRA\, SOCIETY
\end{center}
 
\newpage
 
\begin{center}
{\large {\bf {Suggested Topics for a year long graduate course in\\[2mm]
Theoretical Linear Algebra\\[4mm] }}}
\end{center}
 
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Factorizations\; for\; Rectangular\; and\; Square\; Matrices$}}
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Symmetric \; Matrices\\Positive \; Definite \; Matrices\\
Normal \; and\; Unitary\; Matrices\\ {*}\; Field \; of \; Values
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\hspace*{8mm} Matrix \; Decomposition,\\ \hspace*{3mm} Eigenvalue\; Problem\\
{*}\; Interlacing\; Inequalities\\ {*}\; Majorization\\
{*}\; Perturbation\; Theory
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Commuting\; Matrices\\Functions\; of\; Matrices\\Matrix\; Equations;\\
{*}\; Generalized\; Inverses
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\put(-15,40){Topics for a one semester course,}
\put(-15,35){or the first semester of a year}
\put(-15,30){long course:}
\put(-15,23){Suggestions for quarter systems :}
\put(-15,16){\hspace*{5mm} First quarter :}
\put(-15,8){\hspace*{5mm} Second quarter :}
\put(-15,0){\hspace*{5mm} Third quarter :}
 
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\put(105,28){Topics preceded by an asterix \, $*$}
\put(105,22){are considered optional and}
\put(105,16){should be covered only if desired}
\put(105,10){and time permits.}
 
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\newpage
 
\begin{center}
{\large {\bf {Suggested Topics for a year long graduate course in\\[2mm]
Numerical Linear Algebra\\[8mm] }}}
\end{center}
 
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round-of\!f \; error,\; machine\; constant,\; stability\; and\\ conditioning\;
of\; algorithms,\; vector\; and\; matrix\; norms
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\put(-13,100){\framebox(57,48)[c]{$\begin{array}{c}
LINEAR\; EQU\!ATIONS\; :\\
LU\; factorization\\Gauss \; elimination\\Pivoting\; and\; growth\; factors\\
Cholesky\; decomposition\\Iterative\; methods:\\Iterative\;
refinement,\\Jacobi,\; Gauss-Seidel,\; SOR\\
Sparse\; direct\; methods \end{array}$}}
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QR \;iteration \;with\; implicit\\
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\hspace*{3mm} power\; method\\
Conditioning \;and \;perturbation \\
theory\; of\; eigenvalue\;problems
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Round-of\!f \; errors\; for \;Gauss\\
\hspace*{3mm} elimination \;and\; perturbations\\
Conditioning\; of \;linear\; systems
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Symmetric\;eigenvalue\;problem\\
Generalized \;eigenvalue \;problem\\
QZ\; method
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Large \;scale \;matrices\; :\\
Conjugate \;gradient \;method\\
Lanczos \;and \;Arnoldi \;methods\\
Extremal \;eigenvalues \;and \\
singular \;values
\end{array}$}}
 
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\vspace*{0.5cm}
 
\noindent A one semester course or the first semester of a year long
course should cover topics numbered 1, ... , 4, as
well as actual numerical computations.\\[2mm]
For a quarter system the first quarter would cover topics 1, 2, and 3
combined with actual numerical computations.
The second quarter would cover topics 4, 5, and 6, with the remaining
topics covered in the third quarter.
 
\newpage
 
\begin{center}
{\large {\bf {Suggested Topics for a year long graduate course in\\[2mm]
Applied Linear Algebra\\[12mm] }}}
\end{center}
 
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Matrix\; factorizations:\; LU \; and\; QR \; decomposition,\\
\hspace*{4cm} Singular \; value \; decomposition\\
Canonical\; forms:\; Jordan \;and\; Schur\;
normal \;form
\end{array}$}}
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PERTU\!RBAT\!ION\; TH\!EORY \\
Matrix\; and\; vector\; norms\\
Rayleigh \;principle\\
Courant \;minimax\; theorem\\
Interlacing \;inequalities\\
Hadamard,\; Weyl\; inequalities\\
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NON\! NEGATIVE \;M\! ATRIC\!ES\\
Perron-Frobenius\\
Matrices\; and \; Graphs\\
Laplacian\; matrix\\
Stochastic \;matrices\\
M-matrices\\
Finite\;Markov\;chains\\
Applications \; to\; economics,\; ...
\end{array}$}}
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M\! ATRIX \;EQU\!ATIONS\\
Lyapunov \;equation\\
Sylvester \; equation\\
Ricatti\;equation\\
Theory,\; applications\\
and\; numerics
\end{array}$}}
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SPECIAL\; M\! ATRICES\\
Symmetric \;matrices,\\
Positive\; definite\;matrices\\
Orthogonal,\;unitary\; matrices\\
Spectra\\
Discretization\; matrices\\
Diagonal\; dominance
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Inertia \;and \;Stability :\\
Inertia \;estimates, \;computations\\
Root\;separation\\
Symmetric\;factorization\\
Applications \;and \;computations
\end{array}$}}
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M\! ATRIX \;FU\! NCTIONS\\
Exponential,\; polynomials
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\put(127,20){\framebox(32,22)[c]{$\begin{array}{c}
Field \;of\; values\\
Convexity\\
Location\\
Extensions
\end{array}$}}
 
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\end{center}
 
\noindent A one semester course or the first semester of a year long course
could be made of topics labelled 1, 2,
3, and 4, or alternately of topics 1, 2, 3, and 5.\\[2mm]
For a quarter system, the first quarter could comprise topics 1, 2, 3,
the second could cover topics 4 and 5, with the remainder left for the
third quarter.
 
\end{document}