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\begin{center}
{\bf \Large Bibliography of Random Eigenvalue Literature} 
\end{center}
\begin{center}
Alan Edelman
\end{center}
\begin{center}
Department of Mathematics  \\
University of California \\
Berkeley, CA 94720 \\
\verb+edelman@math.berkeley.edu+
\end{center}
\begin{center}
November 13, 1991
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\vspace{.1in}

Many different
groups are working on random eigenvalues
from many different viewpoints.
The literature is
vast and this bibliography is far from complete.
References \cite{Mu}, \cite{Me}, and \cite{Gi3} are
three books on the subject with very different viewpoints.
\cite{Mu} contains a nice exposition of the multivariate
statistical theory complete with exact distributions using
zonal polynomials.  \cite{Me} contains material of interest
in physics.  \cite{Gi3} contains the nicest theoretical exposition
on the subject, though 
it can be difficult to follow because of  typographical
errors in the mathematics.
 

A more useful classification of this material would 
organize which papers have exact distributions, which have
asymptotic distributions, which contain ``semi-circle" type
laws, and which contain approximations to distributions.
I hope to some day perform this classification.
Please send me any papers on random eigenvalues.




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\begin{thebibliography}{General}

\vspace{.1in}

% GENERAL

\bibitem{CKN}
J.E.~Cohen, H.~Kesten, and C.M.~Newman,  eds.,
{\sl Random Matrices and Their Applications (Contemporary Mathematics 50)},
American Mathematical Society, Providence, 1986.

\end{thebibliography}

\begin{thebibliography}{Numerical Analysis and Related Fields}

\bibitem{Andrew}
A.~Andrew,
Eigenvalues and singular values of certain random matrices.


\bibitem{BG}
G.~Birkhoff and S.~Gulati,
Isotropic distributions of test matrices,
{\sl J. Appl. Math. Physics. 30} (1979), 148--157.

\bibitem{De}
J.~Demmel,
The probability that a numerical analysis problem is difficult,
{\sl Mathematics of Computation 50} (1988), 449--480.

\bibitem{Ed}
A.~Edelman,
Eigenvalues and condition numbers of random matrices,
{\sl SIAM J. Matrix Anal. Appl. 9} (1988), 543--560.

\bibitem{Ed2}
A.~Edelman,
The distribution and moments of the smallest eigenvalue of a random
matrix of Wishart type,
{\sl Linear Alg. Appl. 159} (1991) 55--80.

\bibitem{Ed3}
A.~Edelman,
{\sl Eigenvalues and Condition Numbers of Random Matrices},
PhD thesis, Department of Mathematics, MIT, 1989.

\bibitem{Ed4}
A.~Edelman,
On the distribution of a scaled condition number,
{\sl Mathematics of Computation}, to appear.




\bibitem{Fl}
R.~Fletcher,
Expected conditioning,
{\sl IMA J. Numer. Anal. 5} (1985) 247--273.

\bibitem{Ha}
P.C.~Hansen,
The 2-norm of random matrices,
{\sl J. Comput. Appl. Math. 23} (1988) 1117-120.

\bibitem{Ko}
E.~Kostlan,
Numerical linear algebra and multivariant analysis,  never published.

\bibitem{Sm}
S.~Smale,
On the efficiency of algorithms of analysis,
{\sl Bull. Amer. Math. Soc., 13} (1985), 87--121.

\end{thebibliography}


\begin{thebibliography}{Multivariate Statistics}

% MULTIVARIATE STATISTICS
 
\bibitem{An}
T.W.~Anderson,
{\sl An Introduction to Multivariate Statistical Analysis,}
John Wiley \& Sons, New York,
1958.
 
\bibitem{Bart0}
M.~S.~Bartlett,
On the theory of statistical regression,
{\sl Proc.\ R.\ Soc.\ Edinb. 53} (1933), 260--283.
 
\bibitem{Bi}
C.~Bingham,
An asymptotic expansion for the distribution of
the eigenvalues of a $3$ by $3$ Wishart matrix,
{\sl Ann. Math. Stat. 43} (1972), 1498--1506.

\bibitem{Da}
A.W.~Davis,
On the ratios of the individual latent roots to the trace
of a Wishart matrix,
{\sl J. Mult. Anal. 2} (1972) 440--443.

\bibitem{Ge}
S.~Geman,
A limit theorem for the norm of random matrices,
{\sl Ann. Prob. 8} (1980), 252--261.

\bibitem{Ge}
The spectral radius of large random matrices,
{\sl Ann. Prob. 14} (1986), 1318--1328.


\bibitem{Goodman}
N.R.~Goodman,
Statistical analysis based on a certain multivariate complex
Gaussian distribution, (An introduction),
{\sl Ann. Math. Stat. 34} (1963), 152--177.
 
\bibitem{GuR}
R.D.~Gupta and D.S.P.~Richards,
Hypergeometric functions of scalar matrix argument are expressible
in terms of classical hypergeometric functions,
{\sl SIAM J. Math. Anal. 16} (1985), 852--858.

\bibitem{Ja1}
A.T.~James,
Normal multivariate analysis and the orthogonal group,
{\sl Ann. Math. Stat. 25} (1954), 40--75.

\bibitem{Ja}
A.T.~James,
Distributions of matrix variates and latent roots derived
from normal samples,
{\sl Ann. Math. Stat. 35} (1964), 475--501.

\bibitem{Jo}
D.~Jonsson,
Some limit theorems for the eigenvalues of a sample covariance
matrix,
{\sl J. Multivariate Anal. 12} (1982), 1--38.

\bibitem{KC}
P.R.~Krishnaiah and T.C.~Cheng,
On the exact distributions of the smallest roots of the Wishart
matrix using zonal polynomials,
{\sl Ann.\ I.\ Math.\ Stat.\ 23} (1971), 293--295.

\bibitem{Kr}
P.R.~Krishnaiah,
Some recent developments on real multivariate distributions,
{\sl Develop. Stat. 1} (1978), 135--169. 

\bibitem{Mu}
R.J.~Muirhead,
{\sl Aspects of Multivariate Statistical Theory,}
John Wiley \& Sons, New York,
1982.

\bibitem{OR}
I.~Olkin and S.N.~Roy,
On multivariate distribution theory,
{\sl Ann. Math. Stat. 25} (1954), 329--339.

\bibitem{Pi1}
K.C.S.~Pillai,
On the distribution of the largest characteristic root of
a matrix in multivariate analysis,
{\sl Biometrica 52} (1965), ???.

\bibitem{Pi2}
K.C.S.~Pillai,
Distributions of characteristic roots in multivariate analysis,
{\sl Canadian J. Stats 4} (1976), 157--184.

\bibitem{SKC}
F.J.~Schuurmann, P.R.~Krishnaiah, and A.K.~Chattopadhyay,
On the distributions of the ratios of the extreme roots
to the trace of the Wishart matrix,
{\sl J. Multivariate Anal. 3} (1973), 445--453.

\bibitem{Si}
J.W.~Silverstein,
The smallest eigenvalue of a large-dimensional Wishart matrix,
{\sl Ann. Prob. 13} (1985), 1364--1368.

\bibitem{Su}
T.~Sugiyama,
On the distribution of the largest latent root
of the covariance matrix,
{\sl Annals Math.\ Stat. 38} (1967), 1148--1151.

\bibitem{TC}
C.G.~Troskie and W.J.~Conradie,
The distribution of the ratios of characteristic roots
(condition numbers) and their applications in principal
component or ridge regression,
{\sl Lin. Alg. Appl. 82} (1986), 255-279.  

\bibitem{WCK}
V.B.~Waikar, T.C.~Chang, and P.R.~Krishnaiah,
Exact distributions of a few arbitrary roots of some complex random matrices,
{\sl Austr. J. Statist. 14} (1972), ???--???.



\bibitem{Wi}
S.~Wilks,
{\sl Mathematical Statistics},
John Wiley \& Sons, New York,
1967.

\bibitem{W}
J.~Wishart,
The generalized product moment distribution in samples from a normal
multivariate population,
{\sl Biometrika 20A} (1928), 32--43.

 

\end{thebibliography}
\begin{thebibliography}{Number theory}

% NUMBER THEORY

\bibitem{Od}
A.M.~Odlyzko,
The $10^{20}$-th zero of the Riemann zeta function and
70 million of its neighbors,
to be published.




\end{thebibliography}
\begin{thebibliography}{Computational Complexity, Graph Theory,
and Related fields}

% COMPUTATIONAL COMPLEXITY, GRAPH THEORY and related fields
 
\bibitem{BS}
L.~Blum and M.~Shub,
Evaluating rational functions: infinite precision is finite
cost and tractable on average,
{\sl SIAM J. Comput. 15} (1986), 384--398.
 
\bibitem{Bol}
B.~Bollob\'{a}s,
{\sl Random Graphs,}
Academic Press, London, 1985. (See 347--352).
 
\bibitem{Bo}
R.B.~Boppana,
Eigenvalues and graph bisection: an average-case analysis,
{\sl Proc. 28th IEEE Symposium on Foundations of Computer Science}
(1987), 280--285.

\bibitem{BS}
A.~Broder and E.~Shamir,
On the second eigenvalue of random regular graphs,
{\sl Proc. 28th IEEE Symposium on Foundations of Computer Science}
(1987), 286--294.

\bibitem{FKS}
J.~Friedman, J.~Kahn, and E.~Szemer\'edi,
On the second eigenvalue of random regular graphs,
{\sl Proc. 21st ACM Symposium on Theory of Computing}
(1989).

\bibitem{GS}
U.~Grenander and J.~Silverstein,
Spectral analysis of networks with random topologies,
{\sl SIAM J. Appl. Math. 32(2)} (1977), 499--519.

\bibitem{Mc}
B.D.~McKay,
The expected eigenvalue distribution of a large regular graph,
{\sl Linear Algebra Appl. 40} (1981), 203--216.

\bibitem{newman1}
C. M. Newman,
Lyapunov exponents for some products of random matrices:
exact expressions and asymptotic distributions,
in
J.E.~Cohen, H.~Kesten, and C.M.~Newman,  eds.,
{\sl Random Matrices and Their Applications (Contemporary Mathematics 50)},
American Mathematical Society, Providence, 1986, 121--141.

\bibitem{newman2}
M.Isopi and C.M.Newman,
The triangle law for Lyapunov exponents of large random matrices,
{\sl Commun. Math. Phys. 143} (1992), 591--598.

\bibitem{newman3}
M.Isopi and C.M.Newman,
The distirbution of Lyapunov exponents for large random matrices,
to appear.


 

\end{thebibliography}
\begin{thebibliography}{Physics}
% PHYSICS
 
\bibitem{BFFMPW}
T.A.~Brody, J.~Flores, J.P.~French, P.A.~Mello, A.~Pandey,  and
S.S.M.~Wong,
Random-matrix physics: spectrum and strength fluctuations,
{\sl Rev. Modern Physics 53} (1981), 385--479.

\bibitem{Bronk}
B.V. Bronk,
Accuracy of the semicircle approximation for the density of
eigenvalues of random matrices,
{\sl J. Math. Phys. 5} (1964), 215--220.

\bibitem{Carmeli}
M.~Carmeli,
Statistical theory of energy levels and random matrices in physics,
{\sl J. Stat. Phys. 10} (1974), 259-297. 

\bibitem{Ginibre}
J.~Ginibre,
Statistical ensembles of complex, quaternion and real matrices,
{\sl J. Math. Phys. 6} (1965), 440--449.

\bibitem{JKT}
R.C.~Jones, J.M.~Kosterlitz, and D.J.~Thouless,
The eigenvalue spectrum of a large symmetric random matrix with
a finite mean,
{\sl J. Phys. A11} (1978) L45-L48.

\bibitem{Le}
H.S.~Leff,
Class of ensembles in the statistical theory of energy-level spectra,
{\sl J. Math. Phys. 5} (1964), 763--768.

\bibitem{Me}
M.L.~Mehta,
{\sl Random Matrices,}
Academic Press, New York, 1991.

\bibitem{Po}
C.E.~Porter,
{\sl Statistical Theories of Spectra: Fluctuations,}
Academic Press, New York, 1965.

\bibitem{Wig0}
E.P.~Wigner,
Characteristic vectors of bordered matrices with infinite
dimensions,
{\sl Ann. Math. 62} (1955), 548--564.

\bibitem{Wig1}
E.P.~Wigner,
On the distribution of the roots of certain symmetric matrices,
{\sl Ann. Math. 67} (1958), 325--327.

\bibitem{Wig4}
E.P.~Wigner,
Statistical properties of real symmetric matrices with many dimensions,
{\sl Proc. 4th Can. Math. Congr.} (1959), 174--176.  

\bibitem{Wig3}
E.P.~Wigner,
Distribution laws for the roots of a random Hermitian matrix,
(1962), first published in \cite{Po}, 446.

\bibitem{Wig2}
E.P.~Wigner,
Random matrices in physics,
{\sl SIAM Review 9} (1967), 1--23.


\end{thebibliography}
\begin{thebibliography}{Combinatorics and Probability} 
 
 
% COMBINATORICS and PROBABILITY

\bibitem{Ah}
L.V~Arharov,
Limit theorems for the characteristic roots of a sample
covariance matrix,
{\sl Soviet Math. Dokl. 12} (1971), 1206--1209.

\bibitem{Ar1}
L.~Arnold,
On the asymptotic distribution of the eigenvalues of random matrices,
{\sl J. Math. Anal. Appl. 20} (1967), 262--268.

\bibitem{Ar2}
L.~Arnold,
On Wigner's semicircle law for the eigenvalues of random matrices,
{\sl Z. Wahrsch. Verw. Gebiete 19} (1971), 191--198. 

\bibitem{BhS}
A.T.~Bhaaruchha-Reid and M.~Sambandham,
{\sl Random Polynomials,}
Academic Press, Orlando, 1986.

\bibitem{FK}
Z.~F\"{u}redi and J.~Koml\'os,
The eigenvalues of random symmetric matrices,
{\sl Combinatorica 1} (1981), 233-241.

\bibitem{gi0}
V.L.~Girko,
{\sl Random Matrices},
Vishcha Shkola, Kiev, 1975. (In Russian.)

\bibitem{Gi1}
V.L.~Girko,
General limit theorems for spectral functions of random matrices,
{\sl Theory Prob. Appl. 22} (1977), 156--160.


\bibitem{Gi2}
V.L.~Girko,
Circular law,
{\sl Theory Prob. Appl. 29} (1984), 694--706.

\bibitem{Gi3}
V.L.~Girko,
{\sl Theory of Random Determinants,}
Kluwer Academic Press, Boston, 1990.

\bibitem{Gr}
U.~Grenander and J.W.~Silverstein,
Spectral analysis of networks with random topologies,
{\sl SIAM J. Appl. Math. 32} (1977), 499-519.

\bibitem{Jo}
D.~Jonsson,
Some limit theorems for the eigenvalues of a sample covariance
matrix,
{\sl J. Multivariate Anal. 12} (1982), 1--38.

\bibitem{MP}
V.A.~Mar\v{c}enko and L.A.~Pastur,
Distributions of eigenvalues for some sets of random matrices,
{\sl Math. USSR-Sbornik 1} (1967), 457--483.

\bibitem{Ol}
W.H.~Olson and V.R.R.~Uppuluri,
Asymptotic distribution of eigenvalues of random matrices,
{\sl Sixth Berkeley Symposium on Math. Statist. Probability
3} (1972), 615-644.

\bibitem{Tr}
H.F.~Trotter,
Eigenvalue distributions of large Hermitian matrices;
Wigner's semi-circle law and a theorem of Kac, Murdock, and
Szeg\H{o}, {\sl Advances in Math. 54} (1984), 67--82.

\bibitem{Wa}
K.W.~Wachter,
The strong limits of random matrix spectra for sample matrices
of independent elements,
{\sl Ann. Prob. 6} (1978), 1--18.

\end{thebibliography}

\end{document}
