What isa random matrix volume 52 number 11

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?W H A T I S . . . a Random Matrix? Persi Diaconis

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When I became a professor at Harvard, David Kazhdan was running a "basic notions" seminar--things every graduate student (and perhaps also faculty member) should know. He asked me to give a talk on "what is a random variable." Since a random matrix is a random variable taking values in the space of matrices, perhaps it's good to start with random variables.

I was surprised by Kazhdan's request since "everybody knows" that a random variable is just a measurable function

X() from to X.

He answered "yes, but that's not what it means to people working in probability" and of course he was right. Let us consider the phrase

Pick a random matrix from Haar measure on the orthogonal group On

Here On is the group of n ? n real matrices X with XXT = id . Most of us learn that On has an invariant probability measure ?, that is, a measure on the

Borel sets A of On such that for every set A and matrix M

?(A) = ?(MA), ?(On) = 1.

Let me tell you how to "pick X from ?." To begin

with, you will need a sample Yij of picks from the

standard normal density (bell-shaped curve). This

is

the

measure

on

the

real

line

with

density

. e-x2 /2

2

Even if you don't know what it means to "pick Yij

from the normal density," your computer knows.

You can just push a button and get a stream of in-

dependent normal picks.

Now things are easy. Fill up an empty n ? n

array with Yij , 1 i, j n . Turn this into an

Persi Diaconis is the Mary V. Sunseri Professor of Statistics and Mathematics at Stanford University.

orthogonal matrix by applying the Gram-Schmidt algorithm; make the first row have norm one, take the first row out of the second row and then make this have norm one, and so on. The resulting matrix X is random (because it was based on the random Yij) and orthogonal (because we forced it to be). Using the orthogonal invariance of the normal distribution it is not hard to prove that X has the invariant Haar measure

probability(X A) = ?(A).

Let us now translate the algorithmic description

of a random orthogonal matrix into random variable language. Let = lRn2. Let X be the orthogonal

group. The Gram-Schmidt algorithm gives a map

X() from almost all of onto X. This X() is

our random variable. To prescribe its probability

distribution,

put

product

measure

of

e-x2 /2 2

d

x

on

lRn2.

The push forward of this measure under the map X

is Haar measure ?.

Often, one is interested in the eigenvalues of the

matrix. For orthogonal matrices, these are n points

on the unit circle. Figure 1a shows the eigenvalues

of a random 100 ? 100 orthogonal matrix. While

there is some local variation, the eigenvalues are

very neatly distributed. For contrast, Figure 1b

shows 100 points put down independently at

random on the unit circle. There are holes and

clusters that do not appear in Figure 1a. For details,

applications and a lot of theory supplementing

these observations, see Diaconis (2003).

So far, I have answered the question "what is a

random orthogonal matrix?" For a random unitary

matrix replace the normal distribution on lR with

the normal distribution on C. This has density

e-|z|2

dz.

We

choose

a

random

complex

normal

variable Z on a computer by choosing real

independent normals Y1 and Y2 and setting

NOTICES OF THE AMS

VOLUME 52, NUMBER 11

Figure 1a. Eigenvalues of a random orthogonal matrix.

Z= use

1 2

Y1+ the

i

1 2

Y2

.

qua

For a random symplectic

ternions

and

Z

=

1 4

(Y1

matrix, + iY2+

jY3 + kY4). Random matrices in orthogonal, uni-

tary, and symplectic groups are called the classical

circular ensembles in the physics literature.

There are also noncompact ensembles; to choose

a random Hermitian matrix, fill out an n ? n array

by putting picks from the standard complex nor-

mal above the diagonal, picks from the standard

real normal on the diagonal, and finally filling

below the diagonal by using complex conjugates

of what is above the diagonal. This is called GUE

(the Gaussian unitary ensemble) in the physics

literature because the random matrices have dis-

tribution invariant under multiplication by the

unitary group. There are other useful ensembles

considered in the classical book by Mehta (2004).

One of the interesting claims argued there is that

only three universal families (orthogonal, unitary,

and symplectic) need be considered; many large n

problems have answers the same as for these

families, no matter what probability distribution

governs the matrices involved.

Historically, random matrix theory was started

by statisticians studying correlations between

different features of a population (height, weight,

income...). This led to correlation matrices with

(i, j) entry the correlation between the ith and jth

features. If the data was based on a random sam-

ple from a larger population, these correlation

matrices are random; the study of how the eigen-

values of such samples fluctuate was one of the first

great accomplishments of random matrix theory.

These values and the associated eigenvectors are

a mainstay of a topic called "principle components

analysis". This seeks low-dimensional descriptions

of high-dimensional data; it is widely used across

Figure 1b. Random points on the circle.

applied areas from psychology to oceanography and is a crucial ingredient of search engines such as Google. See Diaconis (2003) for more details.

Physicists began to study random matrix theory in the 1950s as a useful description of energy differences in things like slow neutron scattering. This has grown into an enormous literature which has been developed to study new materials (quantum dots) and parts of string theory. There have also been wonderful applications of random matrix theory in combinatorics and number theory. One can find the literature on all of these topics by browsing in Forrester et al (2003).

Returning finally to random matrices as mathematical objects, the reader will see that we have been treating them as "real" rather than as abstract mathematical objects. From a matrix one passes to the eigenvalues and then perhaps to a spectrum renormalized to have average spacing one and then to the histogram of spacings. All of this is mechanically translatable to the language of measurable functions. However, this is a bit like working in assembly language instead of just naturally programming your Mac. Random matrix theory has evolved as the high order descriptive language of this rich body of results.

References

P. DIACONIS, Patterns in Eigenvalues: The 70th Josiah Willard Gibbs Lecture, Bull. Amer. Math. Soc. 40, 155?178 (2003).

P. FORRESTER, N. SNAITH, and V. VERBAARSCHOT, Introduction Review to Special Issue on Random Matrix Theory, Jour. Physics A: Mathematical and General 36, R1?R10 (2003).

M. MEHTA, Random Matrices, 3rd Edition, Elsevier, Amsterdam. (2004).

DECEMBER 2005

NOTICES OF THE AMS

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