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LSU Historical Dissertations and Theses
Graduate School
1963
Topological Semigroups of Non-Negative
Matrices.
Dennison Robert Brown
Louisiana State University and Agricultural & Mechanical College
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Brown, Dennison Robert, "Topological Semigroups of Non-Negative Matrices." (1963). LSU Historical Dissertations and Theses. 830.
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64—133
BROWN, Dennison Robert, 1934—
TOPOLOGICAL SEMIGROUPS OF NON­
NEGATIVE MATRICES.
Louisiana State University, Ph.D., 1963
Mathematics
University Microfilms, Inc., Ann Arbor, Michigan
TOPOLOGICAL SEMIGROUPS OF
NON-NEGATIVE MATRICES
A Dissertation
Submitted to the Graduate Faculty of the
Louisiana State University and
Agricultural and Mechanical College
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
in
The Department of Mathematics
by
Dennison Robert Brown
B.S., Duke University, 1955
M.S., Louisiana State University, i960
June, 1963
ACKNOWLEDGMENT
It is a pleasure to record my indebtedness to Professor
Robert J„ Koch for his advice and encouragement.
ii
TABLE OF CONTENTS
Page
I
INTRODUCTION
1
II
GENERALITIES
6
III
IV
V
VI
VII
VIII
GROUPS
18
IDEMPOTENTS
25
COMMUTATIVE SEMIGROUPS
34
COMPACT SIMPLESEMIGROUPS
40
BIBLIOGRAPHY
46
BIOGRAPHY
51
iii
ABSTRACT
This paper is devoted primarily to the study of topological
semigroups of non-negative matrices. Usually, these semigroups
are also assumed to be compact.
Theorems on matrices and semigroups, which are germane to the
paper, are first presented.
Attention is focused on the spectrum
of a non-negative matrix.
It is first shown that a compact topological group of non-negative
matrices is finite, by using the spectral properties of these
matrices.
From this theorem it follows that a clan (continuum
semigroup with unit) of non-negative matrices is contractible.
Some results on the existence of I-semigroups in a clan are also
given.
Next, the general structure of non-negative idempotents is
investigated.
As an application of this investigation, the set
of non-negative idempotents of a fixed rank and order is shown
to be arcwise connected.
A similar theorem is obtained for the
subset of stochastic idempotents of fixed rank and order.
Commutative semigroups are next studied.
The Jordan form of
a matrix is used to show that any commutative semigroup of
complex matrices is similar to a triangular complex matrix
semigroup.
This theorem, together with various algebraic and
topological hypotheses, is used to obtain several sets of
sufficient conditions that a semigroup be similar to a semigroup
of diagonal matrices.
The paper terminates with a chapter concerning topological
representations of finite dimensional compact simple semigroups.
It is shown that any such entity
S
in which the idempotents
form a subsemigroup has an iseomorphic imbedding in the non­
negative matrices if and only if the maximal groups of
finite.
S
are
An analogous result is proved in which maximal groups
are Lie groups and complex matrices are used in place of non­
negative real matrices.
simple, if
E
It is also shown that, if
S
is
is connected, and if each maximal group of
totally disconnected, then
E
is a subseraigroup of
v
S .
S
is
INTRODUCTION
Matrices over a .field have appeared throughout the history of
topological algebras, as a source of examples, through
representation theory, and as separate entities worthy of
investigations in themselves.
In the latter category lies the
copious field of locally compact Lie groups, which is among the more
active areas of current mathematical research,,
have not gone unnoticed!
Matrix semigroups
papers by the Russian authors
Suschkewitsh [26; 2?]1 and Gluskin [12; 13; 1^; 153» primarily
algebraic in nature, have appeared in this direction.
Also to be
noted are the papers by Gleason [10°9 11] on one-parameter semigroups,
and the volume of Hille [38].
The latter work contains an extensive
study of normed semigroups.
Moreover, the first investigations of topological matrix semigroups
in the prevailing spirit seem to have been made in an unpublished
paper by M. J. Etter [?] on stochastic matrices.
A matrix is
stochastic if it is non-negative and each r.ow sum is one.
Etter
showed that the stochastic matrices of fixed order form a convex clan,
and studied the structure of stochastic idempotents.
Following
this, Cohen and Collins [5] initiated a study of affine semigroups.
A semigroup is affine if it is a convex subset of a real topological
^Numbers in brackets refer to the bibliography listed at the
end of this dissertation.
1
2
linear space, and the left and right translation mappings are
affine functions»
Among their results is the fact that a finite
dimensional affine semigroup with identity and zero is iseomorphic
to a real matrix semigroup..
They also characterize all one and two
dimensional compact affine semigroups <> Clark [1] has recently
abstracted further contributions in this direction,
Matrix representations of semigroups have been treated in considerable
detail, notably by Suschkewitsh
[25], Clifford [3], and Munn [20J 21],
A complete treatment of the theory of representations of abstract
semigroups in its current form is given by Clifford and Preston [35] •
The lack of topological results in this direction stems primarily
from the loss of the theory of invariant measures;
there is no
theorem about compact topological semigroups analogous to the
Peter-Weyl theorem in compact Lie groups.
Indeed, it will be seen
in this paper (Chapter 3) that several well known semigroups on an
interval cannot occur in a matrix semigroup over the complex numbers.
Specifically, it will be shown that the only I semigroup which can
occur is the interval [0,1]
under real number multiplication.
In the domain of examples, the semigroup of matrices of the form
x
0
y
3.
» x,y e [0,1], x + y < 1 » has been in the folklore of
topological semigroups for some time; it may be imbedded in the plane
as the point set bounded by the
x + y = 1 .
x
and
y
axes and the line
There are several other examples of interest given in
[5], including one example of a compact real matrix semigroup
whose kernel is not convex.
3
This paper is devoted primarily to the study of topological
semigroups of non-negative matrices; some results which are un­
improved by the assumption of no negative entries are also included.
The additional assumption of compactness (in the sense of bicompactness)
is ordinarily also at hand, in order to have available the many
theorems of topological semigroups which require this hypothesis.
Chapter two is divided into two parts; in the first part, definitions
and theorems in topological semigroups required in this paper are
given.
Proofs of these theorems are omitted, but referenced.
The
second part of Chapter two is devoted to the definitions and theorems
concerning matrices which are used in the sequel.
Since this paper
is written from the viewpoint of topological semigroups, rather than
matrices, some details are supplied in this section.
a theorem of Frobenius is proved.
irreducible non-negative matrix
r
M
This theorem states that an
M
which dominates the spectrum of
positive eigenvector.
In particular,
has a non-negative spectral element
M , and to which corresponds a
It is also shown that a non-negative matrix
having a positive eigenvector is similar to the product of a
stochastic matrix and the scalar matrix corresponding to the maximal
real eigenvalue of
M .
These results motivate the cited papers of
Dmitriev, Dynkin, and Kolmogoroff [6]
and Karpelevich [16] in which
the possible eigenvalues of a non-negative matrix are determined. For
reasons to become apparent later, attention is focused on the set of
eigenvalues of modulus one in the case that the maximal real
eigenvalue of a non-negative matrix is also one.
account of the prior results occur in [36].
An expository
A topological space
of
X
onto
X
X
is contractible if the identity mapping
is homotopic to a constant map.
In Chapter three,
the results mentioned above are used to prove that a compact group
of non-negative matrices is finite, from which it follows that a
non-negative matrix clan is contractible.
raised by Wallace [31].
This answers a question
The results concerning I semigroups,
mentioned earlier, are also presented.
Chapter four sets forth structure theorems for non-negative and
stochastic idempotents.
One result of this part is that a rank
non-negative idempotent matrix is the direct sum of
rank one idempotents.
K
K
non-negative
Theorems of this type are then used to show
that the set of stochastic idempotents of a fixed rank is arcwise
connected.
A similar result is proved for non-negative idempotents
of a fixed rank.
The former was conjectured in [7].
The fifth chapter treats commutative semigroups of matrices.
The
principal result of the first part is that a commutative semigroup
of complex matrices is similar to a triangular semigroup.
theorem does not seem to occur in the (modern) literature.
This
The proof
does not depend on topology, but is an exercise in the use of the
Jordan form of a matrix.
In the second part, various sufficient
conditions for a real matrix semigroup to be diagonal are given.
One
result in this chapter not requiring commutativity is the followingJ
let
f
S
be a connected semigroup of real matrices having an identity
and a zero
between
e
and
e , whose ranks differ by one;
f
is also in
then the convex arc
S , and is an I semigroup.
5
The paper concludes with the study of representations of compact
simple semigroups which are finite dimensional.
any such object
S
It is shown that
in which the idempotents form a semigroup, and
the maximal groups are finite, is iseomorphically imbeddable in the
non-negative matrices whose order is a function of the dimension
of
S
and the order of a maximal group of
S .
A similar theorem
is obtained by replacing *'finite" with ’’L ie" and '’non-negative''
by "complex" in the previous sentence.
If the idempotents are
connected, and maximal groups totally disconnected, then the
idempotents are shown to form a semigroup.
It is conjectured by the
author that any finite dimensional compact simple semigroup with
finite groups is iseomorphically imbeddable in the non-negative
matrices of some related order.
The brief mention of cohomology in Chapter three refers to the
cohomology theory of Alexander-Kolmogoroff-Wallace-Spanier.
treatment of this subject can be found in Wallace's invited
address [30].
A brief
GENERALITIES
2.1
Definitions.
S, a Hausdorff topological space, is a
topological semigroup if there exists a function
S
m
on
SKS
which is jointly continuous and associative; that is
m[a,m(b,c)] = m[m(a,b),c].
m(a,b) is written as ab.
multiplication;
In the sequel,
e
m
is suppressed and
Conjunction is also used to indicate set
AB = {ab:a e A,b e B}.
2
idempotent if
An element
e
in
S
= ee = e .
A clan is a compact, connected
topological semigroup having
an element 1such that lx = x =
every
an identity for
S
into
x e S .
1
is a zero for
an element
such that
x e S
is called
S
if
zx = z
is
xl for
S . An element z in
= xz, x e S ;
if
S
has a zero
is nllpotent if there is a positive integer
x^ = z .
A subset
A
of
S
is a left ideal of
SA c A | right ideals are defined dually.
both a left and a right ideal.
A
S
z,
K
if
is an ideal if it is
A (left, right) ideal of
minimal if it properly contains no (left, right) ideal of
S
is
S .
If
S
has a minimal (two-sided) ideal, it is clearly unique; this is false
for one-sided ideals.
called the kernel of
A minimal ideal, if such exists in
S , and is denoted by
K
S , is
without exception.
A topological semigroup containing no proper (left, right) ideals is
(left, right) simple.
A subset
S
is a subsemigroup if
A
function
0(x) = x" \ x e A , is continuous in the relative topology
A .
Finally, if
S,T
A
of
AA c A ;
of
is a subgroup if
A
is algebraically a group, and the
are topological semigroups, then a
6
7
function
for all
f
on
S
into
a,b e S .
additionally,
f
If
T
f
is a homomorphism if
f(ab) = f(a)f(b)
is one to one, it is an isomorphism,, If,
is a homeomorphism, then
f
is an iseomorphism.
Fundamental results concerning the above are due in the main to
y'
Wallace [28], Koch [17], and Numakura [22] ; see the bibliography
in [29]*
Semigroups on sets having no mentioned topology have been
studied by Suschkewitsh [2*0, Rees [23], and Clifford [2; 4].
The
standard work on abstract semigroups is the book of Clifford and
Preston [353*
2.2
Lot
E
The structure
"
denote the set of idempotents of a semigroup
of K .
If
S is a compact semigroup, then
known to contain minimal ideals of all categories, and
a minimal left ideal } = U{RgR
U{H(e):e e K), where
e
.
If
e e K fl.E
H(e)is the maximal subgroup of
.
F
defined on
is an iseomorphism of
fix
e e K
eS fl E
n
H(e)
E .
Then
Se 0 E
H(e)
K
is
K = U{L:L
is
ee K H E
Se = Sb,eS = aS .If
H(e)
onto
S containing
ideal, then L = Se(R = eS)
If a,b e K , then there exists
such that aS D Sb = H(e)= eSe, and
then the function
S
t
is a minimal right ideal) =
L(R) is aminimal left (right)
for some
S .
by
y e eS(Se),
F(x) = xy (F(x) = yx)
H(f), where
y e H(f) „
Finally,
is iseomorphic to the cartesian product
under a multiplication defined in [30].
The
other results above can be found in the references mentioned in the
preceding paragraph.
2._3
I-seraigroups. The topological semigroup structures that an
interval (a homeomorphic image of the interval [0,1]) can support
were among the first characterized.
An I-semigroup (standard thread)
8
is a topological semigroup
S
on an interval such that one endpoint
is an identity and the other a zero0 I-semigroups have been
characterized by Faucett [8 ] and Mostert and Shields [18],
Clifford
has a series of papers on totally ordered semigroups which is
relevant.
It is shown in [18] that the only types existant are the
following;
(i)
S
has the multiplication of the real interval
[0,1] (type 1^).
(ii)
S
has a multiplication isomorphic to the interval
[1/2,1 ]
under the operation
x o y = max{l/2 ,xy}
(type I2 ).
(iii)
S
is idempotent and has a multiplication isomorphic
to the interval [0,1 ] under the operation
x o y = min(x,y} (type 1^).
(iv)
S
is the union of a collection of semigroups of
types I^,I2 »I^
endpoints.
which meet only at their respective
In this case, multiplication of elements
belonging to different semigroups in the union is given
by x o y = min{x,y},
inherited by mapping
that
where the order is that one
S
homeomorphically onto [0,1] such
h(0) = 0, h(l) = 1 .
2.4 Definition.
Let
S
be a topological semigroupcontaining an
idempotent e .
A one-parameter semigroup based at
e
is a
continuous homomorphic image of the additive non-negative real numbers
such that
0
the domain of
(0) = e .
o
A one-parameter subgroup is defined similarly,
" being extended to all real numbers.
Such objects
are local If
2.g
cr" is restricted to a (one-sided) neighborhood of
0 .
Definition. A Lie group is a topological group whose underlying
space is locally euclidean.
This definition is chosen because the
applications of Lie groups in the sequel do not require any
properties of the coordinate transformations mentioned in the standard
definition.
For further information, see [3^3 and [h-1].
In the remainder of this paper, S
unless specified otherwise.
denotes a topological semigroup
The adjective 1'topological** will be
dropped when no confusion seems likely to arise.
be the set of idempotents of
S .
E
will ordinarily
Generally, Roman capital letters
represent sets, and lower case letters elements of sets;
however,
in proofs concerning matrices, it becomes convenient to sometimes
specify matrices by Roman capital letters and their entries, and
associated vectors, by lower case letters.
This ambiguity is
sufficiently restricted to prevent confusion.
The definitions of a matrix, matrix multiplication, and the determinant
of a matrix are omitted.
Matrices are decomposed into block form
and multiplied in this manner without further reference.
blocks are always square submatrices.
Diagonal
For further comment on the
above remarks, and proofs of the elementary theorems involving matrices,
the reader is referred to the works of Albert [333* Halmos [373>
and Wedderburn [^33 •
£L\§ Definitions. The rank of a matrix is the dimension of its rangej
the rank is known to be the number of linearly independent row
10
(column) vectors in any canonical array representing the matrix.
The trace of a matrix is the sum of the elements of its main
diagonal.
A matrix
exists a matrix
M
Q
identity matrix.
is invertible (regular, non-singular) if there
such that
Q
MQ = I = OH ,
is denoted by
where
. A scalar
I
is the
x
is an
eigenvalue (characteristic value, spectral value) of the square matrix
M
if det(M-xl) = 0 . The spectrum
all eigenvalues of
M
with respect
H .
A vector
to x eS(M)
if
S(M) of a matrix is the
v
set of
is an eigenvector of a matrix
M(v) = x(v).
The number of linearly
independent eigenvectors corresponding to a fixed eigenvalue is the
multiplicity of this eigenvalue.
A similarity transformation is any
inner automorphism generated by an invertible matrix
P .
A
permutation matrix is one having exactly one 1 in each row and each
column, and zeros elsewhere.
A row-column permutation, or simply a
permutation, is a similarity transformation generated by a
permutation matrix.
permutation carrying
A matrix
M
M
is reducible if there is a
into the block form
[AO
B C
; otherwise
\
M
is irreducible. The
general linear group of order
formed by the invertible matrices of this order.
is the real numbers, this group is denoted by
field is the complex numbers, by Gl(n,C) .
n
is the group
If the scalar field
Gl(n,R)J
if the
These groups are basic
examples of Lie groups.
The bounded linear transformations on an n-dimensional topological
vector space in a natural manner, with norm defined by
II A H =
^
<aij)2 ) 1^2 » where
A =
*
11
2.7
Theorem [42]. Any locally convex topology on an n-dimensional
topological vector space coincides with the topology of the norm
given above.
2
In particular, the topology of euclidean
n\ n
n
space applied to the
real matrices is locally convex, and is the most convenient
for the considerations of this paper.
In this frame of reference, it
is obvious that matrix multiplication is continuous.
These remarks
indicate a way in which the results of this paper may be related to
normed semigroups.
A fundamental concept in matrices is that of the Jordan form of a
A proof of the following theorem may be found in [36].
matrix.
2.8
Theorem. Let
similar to
M
M
be a complex
n/.n
= diag{ML^,..., M^},
a scalar matrix and
matrix.
= S^
Then
, where
if
2.9
Proof:
Then
I, is the
iC
Let
v
k
kxk
M^ = 1^ +
is similar to diag{ljt>0},
M
for the eigenvalue
Hence
, Mg = 0 + Ng , with
and
Ng
Since
and
x .
x = x^ , and
m' = PMP"1 = diaglM^Mg},
of the type mentioned in 2.8.
follows that
Further,
identity matrix.
be an eigenvector of
Let
H
idempotent
x(v) = M(v) = M^(v) = x^(v).
S(M) c {0,1}.
is
i f } .
Corollary. A rank
where
is
is a nilpotent matrix having zeros and ones
on the first principal subdiagonal and zeros elsewhere.
Sx t S j
M
Ng
M£ = NL
are zero matrices.
where
nilpotent matrices
•4
’ M2 > “
12
2.10
Theorem [3**].
homomorphism of the
2.11
Corollary.
The determinant function is a continuous
n
Let
n
matrices into the scalar field.
M be a matrix, S(M) =
have multiplicity n(i) .
Then
x^}.
Let
det(M) = x ^ ^ x ^ 2 ^...
and trace (M) = n(l)x^ + ... + ^ k ) * ^
.
2.12
fields,
A c B .
and
are similar over
N
Theorem [331.
be
nxn
Let A
and
matrices over
A . If M
B , then they are similar over
Proof:
In applications,
complex numbers.
setting.
Let
B be
N
M
and
A .
A will be the real numbers and
B
the
For simplicity, the theorem is proved in this
M = PNP”^ , P complex .
B,C real, BN = MB, CN = MC .
Then
P = B + iC, with
Hence, for any integer
(B + kC)N = M(B + kC).By examining the Jordan form
easily seen that theremust exist
B + kC
Let
a value of
k
k ,
of P , it is
for which
is invertible.
In the study of non-negative matrices, the theorem of Frobenius
[9]
is fundamental.
as presented in [36].
The proof given here is due to Wielandt [32],
The ordinary inequality symbols, when applied
to real matrices, indicate that every entry of the matrix satisfies
this inequality.
B-A > 0 .
x+
If
If
A =
A
and
let
B are real matrices, A < B' means
A+ = (
).
If
x is a vector,
denotes the non-negative vector obtained in the same manner.
always represents the identity matrix and
space.
V
I
the underlying vector
13
2.13
Lemma.
If
A > 0
is irreducible of order
n , then
(I + A )11”1 > 0.
Proof.
y > 0 , y ^ 0,
It suffices to show that, if
(I + A)(y) has fewer zero coordinates than
y .
(I + A)n“^(y) > 0 , from which it follows that
Without loss of generality, suppose
y =
^uj
then
For this implies
(I + A)n“^ > 0 .
, (I + A)(y) =
|vj ,
/
with
u,v
positive vectors of the same dimension.
with the column dimension of
lv\ =
\o)
since
(I + A) /u\ =
[o!
u > 0, D = 0 .
B
(m \ +
\0j
equal to that of
B C|
D E
/u\
.
loy
Hence
If
A > 0
Proof:
Let
jxj < r .
Note
D(u) = 0 j
r
of
A .
n ,
z
r , unique to within scalar multiples.
C = {x = (x4) eV : (x.) > 0 , x ^ 0}.
A(x) = w = (w.), r
1
X
For
= min{w./x. I x. / 0}, r = max{r
JL
non-negative real numbers by
x
X
Define
f(x) = rx •
defined, it must be shown that
f
Let
2
M
V .
f
on
C
into the
To prove that
is bounded.
2
r
is well
By the continuity of
is continuous
Further, for any positive real number
be the unit sphere of
M ={x = (xi): x^ +...■+ x
I x > 0}.
X
division, the minimum function, and the operator A , f
on the positive cone of
xe C ,
X
r^ = max{t:t(x) < A(x)} .
t , r^x = rx .
A .
If
There is a positive eigenvector
X
let
Then
is irreducible of order
then there exists a positive real eigenvalue
corresponding to
u .
B C
D E
A =
This contradicts the irreducibility of
2.1*4- Theorem (Frobenius).
x e S(A), then
Let
= l).
V
,
In virtue of the preceding remark,
lif
f(C) = f(M n C).
Note that
MflC
N = (I + A)n"1(M fl C).
By 2.13,
positive cone.
f
let
y = (I +
Hence
A)n~\x ) .
it follows that
bounded on
vector
C
Since
A
and
Z = (v e C:f(v) = r}.
Let
u e C , u > 0 .
is continuous on
Vectors in
Z
By irreducibility
r^ > 0 , and therefore
r > 0 .
By 2.13,
r(v) < A(v) ; if
r(x) < A(x) .
A(v) = r(v).
v > 0 .
Now
Therefore
Therefore
t = r
from this fact that
N , there exists
Z
y e V ,y
Z
r
of
A
a
a
A
are titled extremalvectors.
has no zero row;
v e Z , and let
hence
x = (I + A)n“^(v).
r(v) ^ A(v), then
r , this is impossible.
0, t e S(A)
Hence
such that A(y)
is one-dimensional .
If
A
< r
y
y+ e Z J
can have a zero coordinate.
= t(y).
< r.
hence no
It follows
This completes the proof.
A > 0 , then there exists a non-negative
such that
is a non-negative eigenvector
jf
is
. It remains to be
in the above argument, then
Corollary [36].
real eigenvalue
f
z > 0 .
non-zero eigenvector of
Proof;
.
|t| (y+ ) = (ty)+ = (Ay)+ < A(y+ ) ; therefore ' |t|
Note that, if
2.1ft
jf
x = (I + A)n“\ v ) = (1 + r)n“\v), and
Now suppose there exist
Then
Fix
By the maximality of
Hence
< r
A “
(I + A)n-1 > 0 ,
have the desired properties.
Let
x > 0 .
r
such that r = f(z) * r
z
x e M 0 C ,
r (x) < A(x), and
—
f
Let
is a compact subset of the
is bounded onN . For
and, since
r
N
r (y) < A(y) , and thus
z = (z.) e N
1
shown that
is a compact set.
is reducible,
y
|tj
< r
for all t e S(A).
corresponding to
r .
it can be permuted into the form
There
15
/
0
\I
this can be continued until
A
is in block
\A21 A22 /
triangular form
Aij = 0
wl'th each
j > i .
independently;
eigenvalue of
The matrices
hence
A^ ;
A
is nilpotent.
r
for, say,
Let
A^.
an irreducible matrix and
A^
S(A) = U{s(Aii).
may be triangulated
Let
clearly r = max{r^}.
z
Then by filling
Recall that a non-negative matrix A
is one.
Let
A
such that
M
n
be the maximal real
Note that
r > 0
unless
be a positive eigenvector corresponding to
in remaining coordinates with
zeros, a non-negative eigenvector of A
A
ri
is
for
is easily constructed.
stochastic if each row sum of
= (x: there exists an
x e S(A)}.
r
nXn
stochastic matrix
Karpelevich [16] has shown that Mr
is
a curvilinear polygon with roots of unity asvertices, and has set
forth the equations of the bounding curves. Dmitriev, Dynkin,
Kolmogoroff [6] showed that, for
exists
A > 0
r > 0 , the set
with maximal real eigenvalue
r
and
rM = {x: there
n
such that
x e S(A)}.
Applications in this paper (see 3*3) arise in the case
r = 1, | x| = 1 .
where
The foregoing discussion shows that
M is the order of the matrix
chapter is devoted to
2.16
Lemma [6].
Let
an outline of
A = (aij) b®
A.
The remainder of this
the proofofthis fact.
a stochasticmatrix.
x e S(A) , then there exists a convex k-angular polygon
numbers such that
Proof;
Let
belonging to
Q
If
of complex
xQ c Q ,
x e S(A)
x .
x^ = l,k < n ,
Let
, and let z = (z^)
Q
be an eigenvector of
be the convex hull of
{z^}.
Since
A
16
A(z) = x(z), it follows that
= 1 ,
J
xz^
arethe
hence xz^^ e
-Ha a. 4z .,= xz. > i = 1,
j
a-JJ
Q ,i = 1 ... n.
vertices of xQ,clearly
2.17Corollary.
Let
x^ = 1
for some
Proofs
xQ c Q
A, x,
Q, k
n . Now
!
xQ c
Since
thenumbers
Q ,
be as in 2.16.
If
|x ( = 1 , then
j < k .
by 2.16.
Let
z
be a point of maximum modulus in
Q J z
is clearly a vertex of
Q . Since
hence
xz
is also a vertex of
Q .
vertex of
Q , t = 1, ..., k .
Since
|x | = 1 ,
|xz | =
It follows that
Q
has only
x^z
|z | !
is a
k vertices, these
numbers are not all distinct ;
the result now follows.
2.18
is an irreducible non-negative matrix
Theorem [36].
If
A > 0
of order n with maximal real eigenvalue r > 0 , then
A
is
similar
to the product of a stochastic matrix and the scalar matrix rl .
Proof!
By 2.14, there exists a positive eigenvector
corresponding to
invertible.
2.19
r .
Define
Theorem.
Let
Z = r”‘4 >~^AP ;
Let A
real eigenvalue 1.
P = diag(z^, ..., zn).
Let
z = (z^)
Note
computation shows
be a non-negative n x n
x e S(A), )x|
Z
P
is
is stochastic.
matrix with maximal
= 1 . Then
x
= 1
for some
k < n .
Proof:
if
and 2.17 •
a
is irreducible, the proof is immediate on applying 2.18
Otherwise, A
can be permuted into block triangular form
as in 2.16, A = (A^j) with each A ^ irreducible. Let
x e S(A^);
17
since
Jx |
= 1, the maximal real eigenvalue of
Au
must be
1 .
By applying the proof of the previous case, the theorem is proved.
GROUPS
Throughout the remainder of this paper,
nxn
Nn
denotes the set of
non-negative matrices.
The study of compact semigroups in
compact groups in
Nn .
objects are finite.
Nn
begins with the study of
In this chapter it is proved that such
From this it follows that any clan is
Nn
is
contractible.
3 .1
Lemma.
Lot,
complex matrices.
Proofs
Define
x e H(e).
f
11(e) be a compact topological group of n x n
Then
f on
into
Gl(n,C) by
f(x) = x + I - e ,
f(H(e))
is therefore
Gl(n,C), hence a Lie group [4-1].
Therefore
is a Lie group.
3.2
Lemma.
Let
A e S(X), then
Proof:
of
H(e)
is a Lie group.
is clearly an iseomorphism.
a closed subgroup of
H(e)
H(e)
IA {= 1 .
onto the multiplicative group of non-zero complex
numbers, det(G)
is a compact subgroup of the unit circle.
A n|
)detX | =
for each
that
Gl(n,C). If
Since the determinant function is a continuous homomorphism
Gl(n,C)
1 =
X e G , a compact subgroup of
A
i , then |AjJ
=
The group
= 1
Hence
» A i e S<X). Clearly, if ( A J
for each i .
Let
P e Gl(n,C)
< 1
such
= (A-,, Au,..., A).
n
FXP”^
is triangular, diagonal A
PGP”
is compact, therefore contained in a bounded subset
18
jl
19
2
of complex
n
and for each
k, A
k
k
A =
Since diagonal
e PGP
-1
\k
\k
\
, /\ ^ »•••» A n) »
, it follows that, for each
i ,
5 1 » and 3*2 is proved.
IAI^
3.3
space.
Theorem.
1" ■
Let
H(E)
be a compact group in
N_ . Then
n
H(E)
is
finite.
Proof:
H(E)
Since
H(E)
is a Lie group, the identity component
is open [41].
C
It is therefore sufficient to prove that
is totally disconnected.
If
C ^ E , then
C
of
H(E)
has a non-trivial one
parameter group [41], hence elements of infinite order.
The proof is
then completed by contradiction when it is shown that every element of
H(E)
Let
has finite order.
X e H(E).
By2.9
BEB"1 = I 1^ 0 J
B e Gl(n,R) such that
, where rank E is assumed equal to k . Since
°l
\°
BEB'1
and 2.12, there exists
is an identity for
BXB'1, BXB
= IX
0 \ , where
is a
I 0" 0 /
is a rank
k
BHCEjB"1
into
Since
integer
Assume
if
real
k X k matrix#
Gl(n,R)
Let f
be the iseomorphism of
f(BXB"’1) = BXB"1 + I - BEB"1
defined by
.
f(BH(E)B”1 )
m
is iseomorphic to
H(E), it suffices tofindan
T m
t
such that f(BXB~ ) = f(BEB" ) = I .
k < n .
Note
S(X) = SCBXB"1 )
A e SCBXB"1 ), A $ 0 , then det(X
detCfCBXB"1) - A D
A e SCfCBXB"1 )).
A e SCBXB"1 ).
K
= SCfCBXB"1 )) U {o}.For
- /\lj = 0 .
^
= (1 - A ) n~k • det(X,. - AD. )=
ft
ft
Conversely, if
Finally, by 3.2,
Afi 1
and
A
Hence
and
e SCfCBXB"1 )), then
A e SCfCBXB"1')) gives
|A(
= 1 ,
‘
20
therefore
X e N ,
n
\ e S(BXB“^),
j \j = 1
also yields
0
by 2.15 , 1 e SCBXB"1 ) , and
S(f(BXB“^)) c { X • A *
D = PfCBXB"1^ ”1
Note
=
^ < n).
p e Gl(n,C)
such
is lower triangular and diagonal D =
,t.
multiple {t^l )\^ = 1 , t^ < n).
Let
that
..., A n}°
m = least common
m
Then diagonal D = {l,l, ...»
j = i-1, then (Dm^). . = p*(Dm ).. .
1 J
of
k = n ,
In either event,
e S(f(BXB"’'1')), i = 1, ..., n .
Now if
Since
S(BXB"1 )= SCftBXB"1 ) ^ {o} .
By 2.19, SCBXB"1 ) c { \ I /\t = 1, t < n} U {°} . If
a similar argument can be given.
.
l}.
Hence, bythe compactness
i J
Pf(BH(B)B”1 )P’"1,(Dm ). . = 0, j = i-1 .
By a straightforward
i J
induction, it follows that
D
Dl
(D*”) ^ = °» 3 <
=
..., n „
—1
f(BXB“ )has order< m , which
= I , and therefore
Hence
completes the
proof.
Conjecture.
On the separable Hilbert space H
summable sequences, call a bounded linear operator
if it maps the non-negative cone of H into
of real squareA
itself.
non-negative
If
compact topological group of non-negative operators on
the norm topology), then
3j>j5 Corollary.
Let
S
G
G
is a
H (in, say,
is totally disconnected.
be a continuum semigroup in
Nr . Then
K c e .
Proof*.
Fix
e e E n K.
Then
eSe = H(e)(see 2.2). By
3*3, H(e)
is finite.
Since
eSe
is a continuum, it is degenerate, hence
H(e) = e .
Since
K = U{H(e):e e K), the corollary follows.
A topological space is acyclic if
Hn(S) = 0 , n > 0 , assuming reduced
21
groups in dimension zero.
If
S
Clearly contractible spaces are acyclic.
is a clan, it is known [29] that
n > 0 .
Hence
Hn(S) = H^eSe) for
e e K ,
If, also, S c Nn , then by 3.5* Hn(S) = Hn(eSe) = Hn({e}) = 0 .
S
is acyclic.
The stronger result that
S
is contractible
also holds and will be demonstrated later in this chapter.
The following lemma is due to Gluskin [12], and was rediscovered by
the author.
3.6
Lemma.
e, f € E
Proof:
Let S be a
and
If
complex matrix semigroup.
f ^ e , then rank
Suppose rank e = r, e f f .
vev”^ =
I
0
r
0
for
f e eSe .
nxn
f , g
Then
Choose
vfv-1 =
f < rank e .
v
such that
g 0
, since
is an
rxr
complex matrix, and
is an idempotent in Gl(r,C),
f =
2
g
Let
hence
g= I
S be a clan in which,
totally disconnected.
of
1
such that
based at
Proof:
is an
=g . Since
.
= 0. If not,
then
But this implies
e , contrary to assumption.Therefore det(g) = 0
Lemma.
e
0 0I
0
rank vfv-^ = rank f , it suffices to show det(g)
g
Let
for each
and rank f < r .
e e E,H(e)
is
Suppose also that there exists a neighborhood
V n E = {l}.
Then there is an I-semigroup in
S
1 .
It is well known [19] that the existence of the neighborhood
above is sufficient to insure a local one-parameter semigroup
0*-([0,l])
and if
V
in
CT(a) =
V
such that CT(0) = 1,
cT*(b)g, g e H(l), then
cT(a)
a = b
/£H(l), 0 < a < 1 ,
and
g = 1.
In the
V
22
same paper, it is shown that 0“
can be extended to a full
one-parameter semigroup by defining
t e [1,2]
cP(t) =
and proceeding inductively.
0r_([0,oo))
CJ"(1) 0 "( t-1)
for
Now the closure of
is a commutative clan, hence its kernel is a (connected)
group [17], and therefore a single point
z .
It follows by a theorem
of Koch [173 that this clan has exactly 2 idempotents and is an
I-semigroup.
2_._8 Theorem.
Let
S
be a non-degenerate clan in
contains a standard thread from 1
Proof:
to
Nn . Then
S
K .
By 3*6 , there exists a neighborhood
V of
1
containing no
other idempotents ; this follows from the fact that the rank of an
idempotent equals its trace, see 2.9 • By 3«3» each
finite.
Applying 3.7, thereexists an I-semigroup from 1
By 3.6, rank (e) < rank (1).
e
H(e)
eSe
to
is a subclan with unit e .
K , the above argument produces an
rank (f) < rank (e) .
is
I-semigroup from
e
e E.
If
e to f
e E,
After finitely many repetitions of this
procedure, an idempotent
z
of minimal rank in S
isobtained. Clearly
z e K ; otherwise an idempotent of smaller rank can be obtained as
before.
The union of the I-semigroups so obtained is the desired
standard thread.
2.3.
Bemma.
LetS
1 to e e K , andlet
Proof:
Define
F:S
be a clan containing a standard thread T
K c E .Then
T —
> S by
F(x,l) = x , andF(x,e) = exe
S
is contractible.
F(x,t) = t X t .
= e , for each
Then
x e S .
from
23
3.10 Corollary.
*■ 1■■
Let
S
be a clan c N_ .
Then
S
is contractible.
n
By 3.6, it is clear that the I-semigroups composing the standard
thread of 3.8 cannot be semigroups of type 1^ (see 2.3).
The next
lemma is well known and gives more information on this subject.
3.11
Lemma.
Let
there exists
Proofs
3.12
nXn
be a nilpotent
m < n
Corollary.
Then
Let
z
Let
S
complex matrix.
is strictly lower triangular;
be an I-semigroup of complex
Then
hence
nXn
is the union of semigroups of type 1^ .
be the zero of
matrices by
f(z) = 0 .
S
A
nx n
m
A = 0 .
such that
The Jordan form of
matrices.
Proof:
A
S .
f(x) = x-z .
By 3 .6,
S
3.11 , f(S) , and hence
f(S)
Define
f
on
S
into the complex
is iseomorphic to
S , and
contains no subsemigroup of type 1^
J by
S , cannot contain a subsemigroup of type Ig .
The corollary' now follows by the remarks in 2.3 .
For convenience, the results of the latter part of thischapter are
now summarized.
3.13
—
Theorem.
—
— —
(i)
(ii)
Let
S
be a clan,
' S c Nn .
K c E .
There exists a standard thread of at most
n I-semigroups of type 1^
(iii)
Then
S
is contractible.
from 1 to e .« K .
In closing, note that the compactness in 3*3 is essential!
positive
nxn
the
diagonal matrices form a group iseomorphic to the
direct sum of
n copies of the multiplicative group of positive
real numbers.
The non-negativity is also clearly necessary;
the
circle group can be represented iseomorphically by the real matrices
2
.
Jl
IDEMPOTENTS
In this chapter, structure theorems for non-negative and stochastic
idempotents are given.
As an application of these theorems, it is
shown that the set of stochastic idempotents of order
arbitrary rank
k
n
and fixed rank.
*».!
Lemma. Let
E = (©^j) e Nn, E = E ,
e^j
= 0, j=
1, ... n
that
E =
e ^ = 0. Then either
aJi j “ 1, •••, n .
or
Since permutations preserve non-negativity, it may be assumed
e^ = 0 .
ekl = 0 *^
e^
and fixed
is arcwise connected, as well as the set of
non-negative idempotents of order
Proof*
n
Then for each
k = 1, ..., n , either
P0rrauting rows and columns,
^ 0, x 2, ..., t , ©^^_
I E-q
\
, with
e^ = 0
or
it may be assumed that
0, i - t + 1, ..., n . Wrxte
E^ a
t*t submatrix.
The arrangement of
E21 E22
the first row of
rewrite
E
as
E
/0
and non negativity yield
\
F^ 0
, with
»
= 0
j o F 12
0 F
0 F22°
Now
»
22
F31F32 F33/
E2X = [F^-j, F^2], F ^ = E22* and order
and
F^
F22 =
‘ Nate that F12
are, respectively, row and column matrices.
F12F22 = F12,F22
F22 * and
Now
F32 ~ F31F12 + F32F22 + F33F22 *
Multiplying the latter equality on the right by
25
yields
26
F31F12 = 0 *
2 < q < t
or
Since 811 Pair products
Compose the matrix
elq = 0, q = 2,
epieiq » t + 1 < p < n ,
^ ^ 1 2 * e*-^er epi = 0»P = t + 1,
t .
Hence either
F^ = 0
or
F^2 = 0,
and the lemma is proved.
If
E
in 4-.1 is also stochastic, then clearly
rows? hence
e^ = 0
implies
E
cannot have zero
e ^ = 0, j = 1,
n .
This case was
done by Etter [7]*
Lemma.
k.2
(column).
Proof:
If
Let
E = E^ e
, and suppose E
has a positive row
Then rank E = 1.
E
has order 1 , then
E = (1), and the lemma is obvious;
suppose the theorem has been proved for idempotents of order less than
n , and let
E
have order
n .
assumed that the first row of
If
E
By using a permutation, it may be
E
is positive, e ^ > 0,j = 1, ..., n,.
e ^ = 0 , then by ^.1 e ^ = 0, j = 1, ..., n .
/ E^j E^2 ]
may be written as
0
\°
positive diagonal elements.
E . Write
If
E^
E =
E0^ = 0 J
has order
= rank E ^ =
k rows of
IK
r?
'11 ^12
\ E21 E22
Note
possessing all
less than
n , then by
1. Otherwise, it may
e ^ > 0,i = 1 , ..., n .
Assume that the first
of
E^
/
the inductive hypothesis rank E
be assumed that
» with
By a permutation,
for if
using the idempotency,
\
E
*
are all of the positive rows
®il
a
k
k
submatrix .
I
e ^ > 0 , i > k, j < k, then
j =1,
> 0 ,
..., n , which is impossible.
Thus
27
Eu
=
= 4
that
•
sinoe
= 0 .
Bat
*12 = h i h z + h 2 EZ2 - lt fonoHS
B^S^J =
> 0 ; hence
contrary to the previous paragraph. Therefore
irreducible.
By 2.14,
the range of
independent positive vector.
E
E > 0
%zz - 0 ,
, and is
can have only one linearly
Since the canonical unit vectors
are mapped into positive vectors, the dimension of the range of
one; hence
4.3
E
Proof 1
is
has rank 1 .
Corollary. Let
positive row.
E
E = (©^j)
Then all rows of
By 4.2, rank E = 1 .
be a stochastic idempotent having a
E
are identical.
Since the rows of
E
are proportional,
and each row sum is one, these rows must be identical.
Note this
also shows any rank one stochastic matrix is idempotent.
If
E
is a rank
k
idempotent, then
E
is similar to
by 2 .9, and is therefore the direct sum of
The next theorem shows that, if
diagCl^jO)
idempotents of rank 1 .
k
E e Nn , then each summand may also
be chosen to be non-negative.
4.4
Theorem.
If
2
E = E
e Nr , rank E = k , then
E
non-negative idempotents of rank 1 .
sum of
k
Proof;
if e
has rank 1, there is nothing to be proved.
theorem has been proved for idempotents of rank k-1 .
rank k .
Without loss of generality, assume
e_. = 0, i = t + 1, ..., n .
assume
is the direct
t < n . 'Write
If
Suppose the
Let
E
have
e ^ > 0,i = 1,
t = n, then rank E = 1
, with
t
by 4.1;
the order of
t ,
28
.
E ^2 = 0 • Henc®
As in <*.1, it follows immediately that
*11 = EU * ^22 = E22 *
F = I hi
10
’G =
°\
I ^ 1^
E21 = E2A l
F 2 = F, G2 = G , and
FG = 0 .
\
Then E = F + G ,
To get
GF = 0 , note that
this is obtained by multiplying the equality
E21 = E2 l h l + E22E21
since trace F
0
Let
\ E22E21 E22
0 I
E 22E21E11 = 0 *
+ E22E21 *
*11 * ^ /Kl» rank h i
°n th®rieht 17
= traceE ^ ,rank
F = 1, rank G
= k -1.
~ 1*
Thetheorem
now follows from the inductive hypothesis.
^«,5
if
Theorem.
If
E = (e^j)
is a ran^
e ^ > 0 ,i = 1, ..., n , then
E
k
E
is stochastic, then each
Proof:
r
/
k
E^
a rank 1 idempotent.
E^
is stochastic.
Assuming the obvious inductive hypothesis, let
non-negative idempotent.
permuted to the form
/E^
As in the argument of k.k,
0
\
Nn , and
is permutable to the super
diagonal form (E^, ..., Epp} , with each
If
idempotent in
, with
E^
E
be a rank
E
can be
a rank 1 idempotent.
lE2 1 E22 J
Again,
Then
E22E2lEll = 0 .
tE22E21En \ j
Ca4b. .a..
ii ij JJ
and
.
Since
let
- (a^),
= (b^), E22 = (0lJ).
is zer0> and contains a summand of the form
c..,a.. > 0 , b. = 0 .
ii jj
ij
E = diagCE^Egg) •
Therefore
E„, = 0 ,
21
On applying the inductive hypothesis to
, the theorem is proved.
The second conclusion is obvious.
It is known [1] that the set of idempotents of fixed rank over a
29
full real matrix algebra of order n
is arcwise connected.
The
next theorems show that the set of stochastic idempotents of fixed
rank is arcwise connected, as well as the set of non-negative
idempotents of fixed rank.
are utilized in the
Methods
developed earlier
proofs of these theorems.
inthischapter
Denote
byx£ the set
of rank k
stochastic idempotents of order n , and by
of rank k
non-negative idempotents of order n . Clearly
AB = A, BA = B,
X£ (Y^).
The set
This fact
X^
c
.
A,B e X, (Y^) , with either AB = B,BA = A
It is easily seen that if
or
the set
then the convex gull of
A
and
B
lieswithin
will be used in the sequel.
consists of the matrix (1) and is therefore arcwise
connected;
suppose
k < m < n .
Clearly
X™
k
has been proved arcwise connected for
may be assumed less than
n .
X^
will be
proved arcwise connected by a sequence of lemmas which are now
outlined.
Let
Xi = {A e X^ I a.^ = 0}.
imbedded in
X^for each
hypothesis,
each X^
an imbedding of
arcwise connected in
belonging to
in
*k6
xj^ .
1, ..., n .
will be homeomorphically
By use of the inductive
is shown to be arcwise connected.
in
X. .
X^ , each pair
X^,X^.
Next, using
is shown to be
Finally, any idempotent of
x!1
k
not
U X. is shown to be connected to this set by an arc
i j-
This will complete the proof.
Lemma.
—
Proofs
i =
Xk"
Let
For each
n -l
i , X,
”
K
is horaeomorphically imbedded in
A e xj"1 , define f(A) =
Ik 0 \
H
, f(A) e N
.
X.X .
30
»
Decompose
A
•
fA^ A^ \
as
, with
A^^
a square submatrix
A21 A22
of order i - 1 .
PjfCAjP"1 =
Let
be the permutation matrix such that
/An 0 A12^
0
.
Fix
t ^ i , and let
Q
be the
0 0
A21 ° A22
7
invertible matrix such that, for any matrix
M
tH
by the addition of the t —
QP. fCA )?:1 =
I Au
H
th
row to the i— row .
\
, with H = (au
The matrix
QP^ f(A)P“^
with trace equal to trace A = k .
and the mapping defined by
Lemma.
Proof I
Let
... atiJL) ,
n-1
X!
Hence QP.^ f U )?”1 e Xi ,
g^A) = QP^ f(A)P7^
into
is clearly stochastic,
Furthermore,
[QPi f(A)P"1][QPi fCA)?"1] = QPi f(A)P^1 .
4.7
Then
° A22 /
... a^.n ) .
homeomorphism of
is altered from
0 J
\A21
J =
0 A12
M , QM
is a
X. .
x
For each
i , X.
is'arcwise connected.
B s X. .
Decompose
B
as
/
0 B^ \
with
B21 ° B23
\ B31 ° B33/
B.^
a submatrix of order i - 1
column.
Clearly
B
=
j
th
and the zeros representing the i —
D^„ '
B31 B33
e X^
.
Let A = gi(B ) e X^ .
31
Now
BA = B
and
wholly within
AB = A . Hence the line between
Xi .
B
and
A
n X
By the inductive hypothesis, gi(X^“ )
is
arcwise connected, and it has been shown that any element of
n X
can be connected to
)
by an arc in
lies
X^
X^ . Hence the
lemma is proved.
*K8
Lemma.
For
any i,j,X^ and
X^
are connected by an arc in
Xn
\ *
Proof:
It is no loss of generality to assume
n > 3»
for the
rank 1 , order 2 stochastic idempotents are precisely the convex hull
/ 1 0 | and
of the matrices
(1 0 j
connected.
[0 1 ]
,and hence are arcwise
I0 1 )
Under this assumption, fix an integer q,q £ i,j.
\
(
For
n X
A11 A12 |
e *k"l
A21 A22 /
with A ^
h(A) =
a submatrix of order q - 1. Define h:x£~^ — > X^
by
h is clearly a homeomorphic imbedding
^All ° A12 '
0
10
\A21 0 A22
of
in
into
x£-J
x£ .
Letting
to the sets
n X
C. e [X ”
c
iC-i j
anc*
and
X. in
j
x” ,
correspond.
choose
-By the inductive hypothesis, C,
X
connected by an arc in
connected by an arc in
lemma is proved when
n X
J
hence
X^ .
K
Since
q > i,j.
If
h(C^) and
and
e
C
hCCg)
2
are
are
h(C ) e X., h(C ) e X. , the
x
1
^ J
j = n , the argument is similar.
*
32
4.9 Lemma.
Let
A e
be connected to some
Proof;
, a ^ > 0 , i = 1, ..., n .
X^
idempotent.
a.. < 1
ii
.
A,,}, with each
Since
Assume
C e x£
k < n
A.,
can
inQAQ”1 .
JJ
Q
such that
a rank 1 stochastic
is assumed, there is an
a.. e A.,
ii
A
x” .
By 4.5, there exists a permutation matrix
QA.Q~1 = diag{Ain, .
Let
by an arc in
Then
A.,
JJ
i
such that
has order atleast 2.
be the idempotent
diag{A11, ..., A ^ 1
A ..+1 J+1, ... , Akk}, wxth
rank 1 stochasticmatrixof the same
the diagonal entrycorresponding
order as A ^ ,
a
but having
to a ^ , equal to zero.
c^ ,
may
be constructed as a column of ones, different from the column
containing
B
c.. , with
ii*
all other entries zero.
have order less than
one stochastic idempotents.
connected in
n
X , and
iC
arcwise connected in
Since
A., and
JJ
n , they are connected by an arc of rank
Hence
QAQ”'*' and
C
are arcwise
X
C e X. . Consequently, A and
Q” CQ are
l
n
X
Xj^ , and Q“ CQ e X^ for some j . This
completes the proof.
4.10
Theorem.
Proof:
4.11
x£
Apply if.6, 4.?, 4.8, and 4.9.
Corollary.
Proof:
is arcwise connected.
Yk
is arcwise connected.
It is first shown that
property is immediate in
Let
Y^
is arcwise connected;
this
since this set is convex (see 4.3).
A s y J ; recall trace A = 1 .
Assume
akk > 0 . Let
33
aiJc =
i = 1* •••, n .
aii = V k i
> and 1 = A -
Since the rows of A are proportional,
ai i =
Vki
be the rank one idempotent defined by
Then
B = (bij>
b ^ = ^i* ^ij = ^* ^ ^ ^ •
AB = B, BA = A, and therefore the line between
in
.
defined by
Let
C = (c^)
.
c., = 1, i = 1, ..., n , c. . = 0, j
X*C
proves
Therefore
X J
Y^
A
is connected to
and
B
is
B
and
k .Then
C
lies inside
by an arc in
Y^ , which
is arcwise connected.
Y. = {A e y P : a
= O}.
1
k
ii
;
4.6, 4.?, and 4.8
connected.
A
be the rank 1 stochastic idempotent
BC = B, CB = C, and thus the line between
Let
•
are easily
Note
i
X. cz Y. . Lemmas similar to
i
derived to prove
V Y^ is arcwise
Using 4.5 and thearcwise connectedness of
Y^ , a
lemma analogous to 4.9 can be proved to complete the proof of the
corollary.
The arguments establishing these lemmas are omitted
because of their close relation to arguments given in 4.6 - 4.9 .
4.12
Conjectures.
The rank 2, order 3 stochastic idempotents can be
seen to form a simple closed curve.
Does
whose dimension is a function of
and
is
Y^ ?
n
x£
form a manifold
k ? If so, what, topologically,
To what connectedness conditions are the various sets of
non-negative, bounded linear idempotent operators on separable
Hilbert space subject ?
COMMUTATIVE SEMIGROUPS
Commutative matrices have been examined in detail; an excellent
bibliography of pertinent papers is given in [^33*
In this
chapter, using the Jordan form (2.8) of a matrix and a theorem of
Kaplansky, it is first shown that any commutative semigroup of
n
n
complex matrices is similar to a semigroup of lower
triangular matrices (matrices which are zero above the main diagonal).
Following this, commutative semigroups of real matrices are studied.
For a proof of the first theorem, see [^0],
5.1
n
n
Theorem (Kaplansky). Let
S
be a multiplicative semigroup of
matrices over a division ring, consisting of nilpotent
elements.
Then
S
is similar to a semigroup of strictly lower
triangular matrices (matrices which are zero on and above the main
diagonal).
5.2
Lemma.
Let
A
be an n X n
A = diag{A11, ... Akk},
and each
complex
= }\ ±
matrix in Jordan form,
, with each /\ i
a scalar
a lower triangular nilpotent having ones and zeros
on the first principal subdiagonal and zeros elsewhere, and
J
if
i + 3 •
Let
B = (B
decomposed in the same dimensions as
B. . = 0, i ^ j ;
XJ
that is, B
)
ij
be an
A . If
nxn
AB = BA , then
is insuper-diagonal
B — diag{Bjj, ... , Bkk).
3^
complex matrix
form,
35
Proof:
Fix
i = p, j = q .
X, B
+ N B
=
A k pq
pp pq
- N
B
.
pp pq
let
N
Let
= (b. .), i = 1,
pq
= (m..)o
44
By the prior equality,
. j < 5 » and ( A x - A s)bls = 0 • Since
A x / A g » b^g = 0 , and hence
k - 1 rows of B
pq
(A,. - A-)b,
rC
'J
Therefore
5.3
<CS
and
J
= V l V l j
the first
..., r , j = 1, ■>.., s ,
ij
= (n..), N
0*1 -
AB = BA , it follows that
X B
+ B N , whence (A
- \ )B
= B_ N
''q pq
pq qq
x/xp
/xq pq
pq qq
B
rr
Since
b^.. = 0, j = 1, ...» s
.
Assume
have been proved identically zero. Then
= 0 , from which it follows that b, . = 0,j = 1, .
B
pq
Theorem.
matrices, then
Kj
s .
= 0 .
If
S
S
is a commutative semigroup of
nxn
complex
is similar to a semigroup of lower triangular
matrices.
Proof:
The theorem is proved by induction on the order of the matrices
composing
S . If
n = 1, there is nothing to prove.
Suppose the
theorem has been proved for semigroups with matrices of order k < n ,
and let
S
be a commutative semigroup of
If there exists
A e S
eigenvalues, then by $.2
form,
such that
S
S = diag{Sp ..., S^}.
of order less than
triangular matrices.
A
n x n complex matrices.
has
j > 1
distinct
can be decomposed into super diagonal
Each
S^
is a commutative semigroup
n , hence is similar to a semigroup of lower
Since the semigroups
independent of one another, it follows that
semigroup of lower triangular matrices.
S ,S , p / q,
p q
S
are
is similar to a
36
It remains to dispose of the case in which each
eigenvalue. By referring to the Jordan form of
that each
A e S
eigenvalue of
A , itcan be
A ) and a nilpotent; A = A + N •
MN = NM;
A e S,
for, let
e S(A), A =
Let "Yt
elements of
seen
A = A + N, B =
+ M.
Le the semigroup generated by
commute, 7 ^
Let
A + N}.
\o(+ A M + c K N + NM = AB = BA = cxA + A m +
MN = NM.
has a unique
decomposes into the sura of a scalar (the
~ {N 5 for some
then
A e S
If M,N e
,
Then
0 <N + MN,hence
*Y~L •Since
consists of nilpotent elements. Hence,
n » _i
by 5-l» there exists
P
P r i ? , and therefore
such that
is in strictly lower triangular form.
A = J\+ N e S, then
Hence
g .4
PAP-^" = A + PNP-^, which is lower triangular.
PSP-"'" is lower triangular.
Corollary.
having
n
Let
S
A
be as in 5*3°
distinct eigenvalues, then
of diagonal matrices.
and
Finally, if
If there is an
S
is similar to a semigroup
If, furthermore, the matrices in
has a real spectrum, then
S
A eS
S
are real
is similar to a semigroup of
real diagonal matrices.
Proof:
S
The Jordan form of
can be diagonalized.
A
is diagonal; by 5*2 it follows that
The latter part of the corollary follows
from the fact that the Jordan form of
Note that a semigroup of
nxn
A
is real, and (2.12).
diagonal matrices is iseomorphic to
a subsemigroup of the Cartesian product of
field under coordinate multiplication.
n
copies of the scalar
Sufficient conditions,
different from those in 5*^» for a semigroup of complex (real)
37
matrices to be similar to a diagonal semigroup are now investigated,
g .5
Lemma.
Let
(E^, i = 1, ..., k
idempotents such that
be
n
By 3.5» if
by induction on
i < j » then
n .
complex (real)
E^E^ = E^E^ = E^, i < j , rank E^ = r(i) .
Then there exists a complex (real) matrix
Proof:
n
If
P
such that
r(i) < r(j).
The lemma is proved
n = 1, there is nothing to prove.
the lemma has been proved for matrices of order less than
remarked above,
the set (E^).
is the idempotent of maximal rank
r(k)
.
Since
PE^P"^
As
in
E, is an identity
tC
E^ , i = 1, ..., k , it follows that
is some
n .
From 2.9 , there exists a complex (real) matrix P
such that
for
Assume
r(k)Xr(k)
submatrix.
If
, where
r(k) < n ,
then the system
is isomorphic to a system of dimension less than
the lemma is proved, otherwise,
argument is applied to
E^ ^ .
n , and
E^ = IR , in which case the above
Since
r(k - 1) < n , the inductive
hypothesis can be applied to the system {E^}, i = 1, ..., k - 1 , to
complete the proof of the lemma.
5.6
Corollary.
n >< n
Let
S
matrices with zero E^ , rank E^ = s
rank E^ = s + k - 1 .
k
be a commutative semigroup of complex (real)
idempotents (E.}
1
identity E^
Suppose, also, that
as in 5»5» between
En
1
S
contains a system of
and
E,
K
with
38
rank
E^ = s + i - 1 .
Then
S
semigroup of (k - 1 ) X(k - 1 )
Proof:
Let
f
is isomorphic to a diagonal
complex (real) matrices.
be the function defined by
It is clear that
f
is an iseomorphism of
matrices such that rank f(E^) = i - 1 .
such that
f(X)=X-E^,XeS.
PftE^P -1
1 1^
=
0^
0
S
into the
Let
P
Now
PfCSjP"1
n*n
be the matrix of 5*5
has
0
«
identity
I. x 0 j
0
lo'
where
X*
;
Hence if
I iS,
then
Pf(X)P'1 = | X"
J
0
\°
is a (k - 1)X ( k - 1)
submatrix.
Since
X
°
must commute
t
with
I
j, i = 1, ..., k , it follows by direct computation that X
must be diagonal.
The details are omitted.
similarity generated by
Note that, if
S
P
f
and the
are clearly continuous.
satisfies the hypotheses of 5*6 and is, in addition,
compact, then the diagonal entries of
modulus by 1 .
The function
S
are bounded above in
Hence, in the complex case,
the Cartesian product of unit discs)
S
is a subsemigroup of
in the real case,
S
is a
subsemigroup of the Cartesian product of real intervals [-1,1].
S
If
has stochastic entries, the full interval [-1, 1] may still be
realized, as is shown by the
2
2
example
/x
x
5.7
if
then
Corollary.
S
If
has a zero E
S
S
is a semigroup of
and an identity
F
n
n
l - x \ , x e [0,1 ].
1 - x
real matrices, and
whose ranks differ by one,
is iseomorphic to a subsemigroup of the real numbers, and
hence commutative.
If
S
is also connected,then
S
contains the
39
convex arc between
Proof:
E
and F .
In the argument establishing 5*6, the commutativity is not
needed when there are less than 3 idempotents in the chain.
is iseomorphic to a semigroup of
essentially the real numbers.
iseomorphic copy of
Since the function
S
If the ranks of
E
f(x) =
j
x
(-lnx)x
0
x /
S
real matrices, which are
is connected, then the
andthe similarity generated by
must contain
and
not even be diagonal.
If
S
must contain the arc between (1) and (0).
f
are affine mappings,S
1X 1
Hence
F
P
of
5-6
the convex arc
from E
to
differ by more than one,
then S
need
F .
Indeed, the mapping
, x e (0,1], f(0 ) =
j00
|
is an
0 0
iseomorphic imbedding of the unit interval demonstrating this fact.
COMPACT SIMPLE SEMIGROUPS
The theorems in this chapter are involved with topological
representations of finite dimensional compact simple semigroups,
for a thorough treatment concerning representations of algebraic
semigroups, see [353*
The characterization of compact simple /
semigroups mentioned in 2.2 and treated in [30] is assumed without
further reference.
6.1
Theorem.
Let
S
be a compact, simple, idempotent semigroup
contained in euclidean n-space.
imbeddable in
Proof:
N_ ._
2n+2
En
S
is iseomorphically
.
By the compactness,
homeomorphism of
Then
S
carrying
is bounded.
S
Hence there exists a
into the non-negative cone of En .
Assume this has been done.
For each
to be the (n + l ) X ( n + 1)
non-negative real matrix having
1» yx, •••» yn
as its first row and zeros elsewhere.
B(y)
be the transpose of A(y).
B(y)
are idempotents.
Fix
.
0
xS , since
y = (y^) e S , define
Note, for every
x e S .
f(xS)
For
A(y)
Further, let
y e S,
A(y) and
y e xS , let
is clearly a homeomorphic image of
B(x)
A
right trivial;
is a homeomorphism.
that is
Now, multiplication in
yz = z, zy = y
A(y) A(z) = A(z) , A(z) A(y) = A(y), and
follows that
f(y) f(z) = f(z)
and
40
for all
B(x)
xS
y,z e xS .
is
Since
is an idempotent, it
f(z) f(y) = f(y) .
Hence
f
a
maps
xS
f(z) =
iseomorphically into
0
xS
defining
f
is a
B(z)
/
homeomorphism of
on
^
, z e Sx, it is easily seen that
/ A(x) 0
\
^2n+Z *
and
Sx
Sx U xS
into
separately.
»
which is an isomorphism
It remains to extend
f
over all of
S .
Suppose
v / xS U Sx .
written uniquely as
Define
Since
zy , with
S = SxS = (Sx)(xS),
z = Sx fl wS
/A(y) 0
f(w) = f(z) f(y) =
uniquely as
That
f
is represented
/
zy, z e xS, y c Sx, it follows that
S
have different coordinates.
suffices to show that
f
be a sequence in
f
is well-defined.
are compact;
S
being compact, it
is continuous to complete the proof.
S
converging to
be represented uniquely as
xS, Sx
w
is one to one is immediate from the fact that different
elements of
{wn )
can be
y = xS 0 Sw .
Since
^
B(z)
\
and
w
z y , with
n n
w = zy .
Each
z e Sx,y„ e xS.
n
n
Let
wfi may
Now,
therefore there exist convergent subsequences
^yn(i)^ ^zn(i)^ * having the same indices, such that
(y^^)}
converges to
By continuity
a e xS , (zn^ ) ) converges to
of multiplication in
S , {zn (i)yn(i)}
b e Sx .
converges to
w = ba, and by the uniqueness of representation,
Hence
f
^zn^ j ) converges to
z
and
{y^^}
has been shown to be a homeomorphism on
f(zn(i))
converges to
f(z)
and
f(yn(jj)
ba .
b = z
and
converges to
Sx U xS .
to
Hence
a = y .
y .
Now,
Therefore
f(y) .
Since
f(wn(i)> = f(V D yn(i)) = ' ' V D 1 f(yn(i)) ’ and
f(w) = f(zy) = f(z) f(y), and matrix multiplication is continuous, it
kz
follows that
f(wn(.jj) converges to
f(w), and
f
is continuous.
Note that uniqueness of representation is essential to this proof.
6.2
Corollary.
If
S
is a compact, simple, idempotent,
n-dimensional semigroup, thenS
Proof:
is iseomorphically imbeddable
It is well known [39] that an n-dimensional space is
topologically imbeddable in
6.3
Corollary.
that
Let
S
.
be a compact simple semigroup in E11such
E ,the idempotents of
(i)
S
S , form a semigroup.
(ii)
S
S
is finite ;
S
is a Lie group.
is imbeddable in the non-negative matrices, then each
maximal group is finite by 3*3*
6.1 .
S
is iseomorphic to a complex matrix semigroup if,
and only if, each maximal group of
If
Then:
is iseomorphic to a non-negative matrix semigroup
if, and only if, each maximal group of
Proof:
Since
H(x)
Conversely, fix
x e E
as in
is finite, it is isomorphic to a subgroup of
, the permutation group on
k
elements, for some
.
to the group of
(see 2.6).
be the composite isomorphism imbedding
If
Let
in P^ .
g
Since
t e H(x) , define
H(x)
is finite, g
h(t) =
y e xS
n
E .
E
defined in
Then
, in
permutation matrices P^
is trivially continuous.
/ f(x) 0
\ 0
function on
kxk
k
turn, is isomorphic
H(x)
in
, where
f
is the
g(t);
6.1. To extend
h
H(y) = H(x)y , and this
to
xS, let
right translationmapping
k3
is an iseomorphism.
Let
h(y) =
.If
f(y) 0
0
gU)
s = ty, t € H(x), and this representation of
. The
of
xS
To extend
h
Sx
S , let
w eE,
0 E fl Sw .
is clearly an iseomorphism
h(p) = h(z)h(t)h(y) .
p , the extension of
2n + 2 + k .
h
h
w /£ xS U Sx .
Let
Then zH(x)y = H(w), and the
mapping this defined isan iseomorphism.
of
is unique. Define
in the same manner as in 6.1 .
over all of
z = Sx n E n wS, y = xS
Define
h
s
into the non-negative matrices of dimension
is now extendable to
s e H(y) , then
Let
p e H(w), p = zty,t e H(x).
By the uniqueness of this representation
is well defined.
Now if
p,q e H(w), p = zty, q = zsy, s,t e H(x), then h(p)h(q) =
h(z)h(t)h(y)h(z)h(s)h(y) .
Since
E
is a semigroup, yz = z J
hence
h(y)h(x) = h(yz) = h(x) , and the above expression collapses to
h(z)h(ts)h(y) = h(ztsy) = h(pq), so that
S
is imbeddable in
each maximal group of
The
S
the complex matrices of some dimension, then
is iseomorphic to a compact group of complex
matrices, hence is a Lie group (3*1
H(x)
is a homomorphism.
h are developed exactly as in 6.1 .
remaining properties of
If
h
is a compact Lie group.
and
£3*0)*
Then there exists a faithful
representation, that is an iseomorphism, of
On replacing the mapping
g
Conversely, suppose
H(x)
into
Gl(k,C) [3*0*
of the previous paragraph by the faithful
representation mentioned above, it is easily seen that the analogous
definition of
h
yields the desired imbedding.
Note that right (left) simple compact semigroups in
En
with the
proper types of maximal groups satisfy the hypotheses of 5*3> and
hence have topological representations in the complex matrices.
6.4
Lemma.
H(e)
2
S
be a simple semigroup with
totally disconnected.
Proof:
C
Let
Let
G
Then
E
be the component of
is connected and meets
is a subsemigroup.
Econnected, each
is a subsemigroup of
S
containing
C , it follows that
It remains to show
H(e) D C = eSe fl C = eCe = {e}, since
E . Since
2
C
C = E . Let
eCe
S .
c C , hence
e e E .
is connected.
C
Then
This
completes the proof.
6.5
Corollary.
semigroup with
Let
E
S
be a finite-dimensional compact simple
connected, each
H(e)
finite.
Then
S
is
iseomorphically imbeddable in the non-negative matrices of some order.
6.6
Con.jecture.
Any finite dimensional compact simple semigroup
with finite maximal groups is imbeddable in the non-negative matrices
of some order.
6.7
Theorem.
Let
S
be a compact simple semigroup in
contain a diagonal idempotent.
Proof:
I I, 0 \
E
is a semigroup.
By a row-column permutation, it may be assumed that
1° ”)
If
Then
Nn . Let
e S , with
k
the rank of
S . Call this element
f e eS D E , g e Se fl E , then by computation, f = /
k
0
e .
F \ ,
0
S
45
g =
/
0\
, with
F, G
non-negative matrices of dimensions
iG °i
k x ( n - k) and (n - k ) x k
fg =
respectively.
/Ik + FG 0 | e H(e).
If
Now
(fg)11 = e , then
(I,, + FG)t = It
0
By non-negativity, FG = 0 , hence fg = e . It follows that the
*
product of any pair of idempotents is an idempotent, hence E is
a semigroup.
The idempotents of a simple semigroup of non-negative matrices need
not form a semigroup.
However, any counterexample must have at least
two distinct minimal left ideals, two distinct minimal right ideals,
and groups of order at least two, a total of at least eight elements.
The following example is one of this type.
6.8
E =
Example.
[I J
Let
, F =
0 0 I
X = I J I
Let
with
,11
'1 0
, I =
J J
, J =
'
Ol
i.
i
0 1 ]
11 0
, G = ,00
,
loo I
0 0
Then
I =
K =
I1 1 /
, Z =
0 0
0 0
Let
.00
^J 1
w
I J
H(E) = {E,X}, H(F) = {F,Y}, H(G) = {G,W}, H(K) = {K,Z} .
S = H(E) U H(F) U H(G) U H(K) .
ES = FS, SE = SK .
idempotents of
S
of order less than
Furthermore,
Then
S
is a simple semigroup
FK = X , which shows the
do not form a semigroup.
For non-negative matrices
it may be shown by computation that the
idempotents must form a semigroup.
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XXXIV(1933), 865-871.
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_______________ .
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LXIV(1942), 327-3^2.
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_______________ .
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5.
Cohen, Haskell and Collins, H. S.
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Dmitriev, N., Dynkin, E.', and Kolmogoroff, A.
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characteristic roots of stochastic matrices” ,
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Etter, M. J. P.
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Faucett, W. M.
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Frobenius, G.
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Gluskin, L. M.
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Akademii Nauk SSSR, XXII(1958), 439-448.
13 .
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Matrices''4 Doklady Akademii Nauk SSSR,
XCVII( 195*0, 17-20.
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semigroups of matrix algebras", Uspehi
Matematicheskih Nauk, XI(1956), 199-206.
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_______________ . '' Semigroups of homeomorphic mappings
of a segment'', Matematicheskii Sbornik.
IXL(1959), 13-28.
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Karpelevich, F. I.
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with non-negative elements", Izvestia Akademii
Nauk. XVI(1951)» 361-383.
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Koch, R. J.
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University Dissertation, 1953*
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Mostert, P. S. and Shields, A. L.
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of semigroups on a compact manifold with boundary'',
Annals of Mathematics. LXV(1957)> 117-143*
19. _______________ .
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Society, XCVI(1960), 510-517-
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Munn, W. D.
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Suschketwictsch, A.
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Wallace, A. D.
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1+8
30. ______________
.
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National Academy of Science, XLII(1956)> 430-432.
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, "Problems concerning semigroups", Bulletin
of the American Mathematical Society. LXVIII(1962),
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Wielandt, H.
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33*
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50
BIOGRAPHY
Dennison Robert Brown was born in New Orleans, Louisiana, on
May 17, 193^.
He was educated in the public schools of New
Orleans, Lake Charles, and Baton Rouge, Louisiana, and of
Winnetka, Illinois.
He attended Duke University upon
graduation from high school, receiving the B. S. degree in
physios fron that institution in 1955*
From June of that year
until May, 1958* he served in the United States Navy, and
currently holds the rank of Lieutenant, USNR.
In June, 1958, he married Janet Madden of Baton Rouge.
They
are parents of one son, Robert, born in October, 1957.
From September, 1958» to June, 1961, Mr. Brown was a member
of the Mathematics staff at Louisiana State University in New
Orleans.
He received the Master of Science degree from
Louisiana State University in i960, and is currently a candidate
for the Doctor of Philosophy degree in Mathematics.
51
EXAMINATION AND THESIS REPOET
Candidate:
Dennison Robert Brown
Major Field:
Mathematics
nritlG of Thesis i
Topological Semigroups of Non-negative Matrices
Approved:
Q
)^rdL.______
Major Professor and Chairman
~^ y i ^ p c-7
Dean of the Graduate School
EDLAMENTING COMMITTEE:
m
(7!
___
tj.
Date of Examination:
May 10, 1963
—